lol  -«  ' 


THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


tamp*"4        '   w 


-v-fr-v-frv-t-y^^j-v-i-y^r^-C^Y^i-y^ 


educational 


EDITED  BY  HENRY  SUZZALLO 

PROFESSOR   OF  THE  PHILOSOPHY    OF   EDUCATION 
TEACHERS  COLLEGE,   COLUMBIA    UNIVERSITY 


THE  TEACHING  OF 
PRIMARY  ARITHMETIC 

A  critical  study  of  recent  tendencies  in  method 

BY 
HENRY   SUZZALLO 


WITH  AN  INTRODUCTION  BY  DAVID  EUGENE  SMITH 


HOUGHTON  MIFFLIN  COMPANY 

BOSTON,  NEW  YORK  AND  CHICAGO 

fiitiettfbe  ptcjsrf  Cambridge 


COPYRIGHT,   1911,  BY  TEACHERS    COLLEGE,  COLUMBIA  UNIVERSITY 
COPYRIGHT,  1913,  BY  HOUGHTON  MIFFLIN  COMPANY 

ALL  RIGHTS  RESERVED 


•"V 


A 

5 


CONTENTS 

INTRODUCTION  .    .     David  Eugene  Smith      v 

THE  TEACHING  OF  PRIMARY  ARITH- 
METIC .......     Henry  Suzzallo 

I.   THE  SCOPE  OF  THE  STUDY  .....      i 
II.  THE  INFLUENCE  OF  AIMS  ON  TEACHING      9 

III.  THE  EFFECT  OF  THE  CHANGING  STATUS 

OF  TEACHING  METHOD  .....    21 

IV.  METHOD  AS  AFFECTED  BY  THE  DISTRI- 

BUTION AND  ARRANGEMENT  OF  ARITH- 
METICAL WORK    ........    32 

V.  THE  DISTRIBUTION  OF  OBJECTIVE  WORK    42 

VI.  THE  MATERIALS  OF  OBJECTIVE  TEACH- 

ING   ............    47 

VII.   SOME  RECENT  INFLUENCES  ON  OBJEC- 

TIVE TEACHING    ........    53 

VIII.  THE  USE  OF  METHODS  OF  RATIONALI- 

ZATION  ...........    60 

iii 


CONTENTS 

IX.   SPECIAL  METHODS  FOR  OBTAINING  AC- 
CURACY, INDEPENDENCE,  AND  SPEED  .    69 

X.  THE  USE  OF  SPECIAL  ALGORISMS,  ORAL 

FORMS,  AND  WRITTEN  ARRANGEMENTS    83 

XI.   EXAMPLES  AND  PROBLEMS 96 

XII.   CHARACTERISTIC    MODES    OF    PROGRESS 

IN  TEACHING  METHOD no 

OUTLINE .119 


INTRODUCTION 

J"    '    "'     ' '  •>  - 

BY  DAVID  EUGLiNb    SivilTH 

2^  ^.  -2.  a  a 

THE  evolution  of  the  teaching  of  primary  arith- 
metic extends  over  a  period  of  about  two  hundred 
years,  although  numerous  sporadic  efforts  at 
teaching  the  science  of  number  to  young  child- 
ren had  been  made  long  before  the  founding 
of  the  Francke  Institute  at  Halle.  During  the 
eighteenth  century  not  much  progress  was  made 
until  there  was  established  the  Philanthropin  at 
Dessau,  and  perhaps  it  would  be  more  just  to 
speak  of  primary  arithmetic  as  having  its  real  be- 
ginning in  this  institution  at  about  the  time  that 
our  country  was  establishing  its  independent  ex- 
istence. It  is,  however,  to  Pestalozzi,  at  the  be- 
ginning of  the  nineteenth  century,  that  we 
usually  and  rightly  assign  the  first  sympathetic 
movement  in  this  direction,  and  it  is  the  period 
from  that  time  to  the  present  that  has  seen  the 
real  evolution  of  the  teaching  of  arithmetic  to 
children  in  the  first  school  years. 
v 


INTRODUCTION 

The  evolution  of  this  phase  of  education  is  one 
of  the  most  interesting  and  profitable  studies 
that  a  teacher  of  arithmetic  can  undertake.  It  re- 
veals the  experiments,  many  of  them  puerile  but 
a  few  of  them  virile,  that  have  been  made  from 
time  to  time ;  it  brings  into  light  the  failures 
which  should  serve  as  warnings,  and  the  suc- 
cesses which  will  inspire  the  teacher  to  better 
and  more  carefully  considered  effort ;  it  shows  the 
trend  of  primary  instruction,  and  it  makes  the 
student  of  education  more  sympathetic  with  the 
great  problem  before  him.yjln_particular,  he  will 
see  the  danger  of  narrowness  in  matters  of 
method,  the  futility  of  expecting  to  create  genu- 
ine interest  by  any  single  line  of  devices,  the 
pmfnt-fesijks  that  came  from  over-emphasis  of 
the  doctrine  of  formal  discipline,  and  the  errors 
of  judgment  that  have  been  made  in  deciding 
upon  what  constitutes  reality  in  an  arithmetical 
problem  for  children.  If  by  such  a  study  he 
should  see  the  childishness  of  confining  one's 
self  to  the  use  of  cubes  alone,  or  of  some  special 
type  of  number  chart,  or  of  some  particular  form 
of  number  cards,  or  of  sticks  of  varied  lengths  or 
vi 


INTRODUCTION 

shapes,  the  result  would  be  salutary.  If  by  his 
study  of  the  absurd  extreme  to  which  formal  dis- 
cipline was  carried  a  generation  ago  he  is  led  to 
see  the  equal  absurdity  to  which  so  many  teach- 
ers are  tending  to-day  —  the  denying  that  any 
such  discipline  exists  at  all  —  his  labor  will  bear 
good  fruit  in  the  school  room  of  the  present. 

It  fortunately  happens  that  after  this  century 
of  experiment  we  are  getting  about  ready  to  take 
some  account  of  stock ;  to  weigh  up  values  ;  to 
select  from  what  the  world  has  produced,  and  to 
select  with  some  approach  to  good  judgment.  It 
is  not  probable  that  the  time  is  entirely  ripe  for 
this  labor,  because  we  are  at  present  in  the  midst 
of  a  period  of  agitation  that  seems  certain  to  warp 
our  judgment,  the  period  of  agitation  fora  some- 
what narrow  phase  of  industrial  education ;  but 
even  with  this  danger  we  are  better  able  to 
weigh  up  the  values  in  the  teaching  of  arith- 
metic than  we  have  ever  been  before.  One  rea- 
son for  this  is  that  we  now  have  men  and  women 
of  sufficiently  sound  education  and  sufficiently 
broad  view  to  attack  the  problem.  These  men 
appreciate  the  efforts  of  Pestalozzi,  but  they 
vii 


INTRODUCTION 

recognize  that  these  are  to  the  present  what 
the  science  of  Franklin  is  to  that  of  Kelvin  and 
Thompson.  They  appreciate  what  Tillich  did  for 
number  work,  and  the  influence  of  Grube  ;  but 
they  know  that  these  men  were  narrow  in  view 
and  dogmatic  in  statement,  and  that  they  stand 
rather  as  warnings  than  as  founders  of  any 
worthy  theory. 

In  looking  for  such  a  man  to  prepare  a  report 
upon  the  teaching  of  arithmetic  in  the  primary 
grades,  the  American  members  of  the  Interna- 
tional Commission  on  the  Teaching  of  Mathe- 
matics turned  first  of  all  to  Professor  Suzzallo.1 
They  felt  that  his  standing  as  a  scholar,  his  ex- 
perience as  a  practical  school  man,  and  his  posi- 
tion in  the  educational  world  fitted  him  perfectly 
for  a  work  of  this  importance.  That  their  good 
judgment  did  not  fail  them  will  be  seen  in  read- 
ing the  following  report.  In  it  Professor  Suzzallo 

1  The  material  presented  in  this  study  was  originally  col- 
lected and  organized  for  the  purposes  of  a  special  report  to 
a  sub-committee  of  the  International  Commission  on  the 
Teaching  of  Mathematics.  This  commission  was  created  at  the 
International  Congress  of  Mathematicians  held  in  Rome  in 
1908.  The  report  was  first  published  in  the  Teachers  College 
Record,  March,  1911. 

viii 


INTRODUCTION 

has  set  forth  very  clearly  the  aims  of  instruction 
in  primary  arithmetic,  rightly  considering  these 
aims  in  their  evolutiojg.a£%  rather  than.  in  their 
static  aspec^s^.  emphasizing  the  importance  of 
this  phase  of  the  study, 


tendencies  that  seem  makingjorji_more  rational 


view  of  teaching.  He  has  discarded  the  narrow 
and  trivial  concept  of  method  that  characterized 
the  educational  work  of  a  generation  now  passing 
away,  and  has  brought  dignity  to  the  term  by 
considering  it  from  the  modern  scientific  stand- 
point. He  has  discussed  both  historically  and 
psychologically  the  important  question  of  object 
teaching,  showing  the  failures  that  have  resulted 
from  narrow  views  of  the  purpose  of  such  an 
aid  and  the  success  that  may  be  expected  from 
a  more  rational  use  of  number  material.  The 
whole  question  of  the  role  of  reason,  or  rather  of 
the  pupil's  effort  to  express  the  rationalizing  pro- 
cess has  been  considered,  the  extreme  danger 
points  have  been  indicated,  and  the  bearing  of 
some  of  our  saner  forms  of  psychology  upon  the 
subject  has  been  set  forth.  The  technique  of  num- 
ber, including  the  question  of  accuracy  and  speed 
ix 


INTRODUCTION 

in  the  operations,  has  also  been  treated  in  a  very 
acceptable  manner,  Professor  Suzzallo's  experi- 
ence in  this  phase  of  work  having  been  unusual. 
And  finally,  the  vexed  question  of  what  consti- 
tutes a  genuinely  concrete  problem  has  been  con- 
sidered in  its  various  bearings,  and  the  present 
tendencies  in  problem-making  have  been  indi- 
cated. It  is  needless  to  say  that  the  last  word  has 
not  been  spoken  on  this  phase  of  the  work,  and 
that  it  never  will  be.  New  generations  produce 
new  lines  of  application  of  arithmetic,  even  for 
children.  But  with  respect  to  the  general  prin- 
ciples of  the  selection  of  matter  for  the  framing 
of  problems,  Professor  Suzzallo  speaks  with  a  con- 
viction that  will  carry  weight. 

With  all  of  the  opinions  expressed  in  such  a 
report  probably  no  reader  will  agree.  It  would  be 
a  poor  discussion  that  would  not  provoke  some 
opposition.  But  with  the  general  tenor  of  the 
report  it  is  certain  that  most  thinking  teachers 
of  to-day  will  find  themselves  in  hearty  accord. 
More  important  still  is  the  fact  that  the  report 
sets  forth  in  clear  language  the  present  status  of 
primary  arithmetic  in  the  more  thoughtful  edu- 


INTRODUCTION 

cational  circles  in  this  country,  and  that  it  will 
state  to  teachers  at  home  and  abroad  the  tend- 
encies as  they  now  appear  to  the  leaders  of  edu- 
cational thought  in  the  United  States. 

TEACHERS  COLLEGE,  COLUMBIA  UNIVERSITY, 
February,  1911. 


THE  SCOPE  OF  THE  STUDY 

IT  is  the  function  of  this  study  to  convey  some 
notion  of  the  methods  employed  in  teaching  math- 
ematics in  the  first  six  grades  of  the  American 
elementary  school.  No  attempt  is  made  to  give  a 
minute  description  of  the  endless  details  of 
teaching  procedure,  nor  even  to  enumerate  all 
the  types  of  teaching  method  employed.  Its 
purpose  is  restricted  to  an  analysis  of  the  larger 
tendencies  in  teaching  practice  which  are  repre- 
sentative of  the  spirit  of  mathematical  instruction 
in  the  lower  schools. 

Function  of  the  Study  to  Trace  General  Tendencies 

We  should  have  a  much  simpler  task  if  it  were 
ours  to  sketch  the  purposes  of  mathematical  iji- 
struction,  or  to  outline  the  nature  and  organiza- 
tion of  the  various  mathematical  courses  of  study. 
As  it  is,  we  have  to  describe  something  less  con- 
cr^tc,  namely  the  method  employed  in  the  pre- 

9  I 


TEACHING   PRIMARY  ARITHMETIC 

sentation  of  mathematical  subjects.  Intimately 
dependent  upon  the  subject  matter  involved, 
dominated  by  the  special  aims  of  mathematical 
instruction  in  a  non-technical  school,  adjusted  to 
the  immaturities  of  childhood,  and  reflecting  the 
personal  habits  of  mind  of  the  teacher,  —  teach- 
ing method  emerges  —  a  powerful,  variable,  and 
subtle  thing.  In  spite  of  subtlety  and  variability 
there  are,  however,  certain  general  practices 
that  can  be  described  and  analyzed.  It  is  with 
these  that  this  study  will  deal 

Teaching  Method  is  a  Mode  of  Presentation 

Owing  to  the  existing  confusions,  it  is  well  at 
the  very  outset  to  have  in  mind  a  clear  definition 
of  the  term  "  teaching  methods."  Teaching 
methods  are  always  methods  of  presentation.  In 
this  respect  the  teaching  art  is  like  any  other 
art,  literary,  graphic,  plastic,  or  what  not.  The 
literary  artist,  for  example,  has  a  purpose,  a  sub- 
ject matter,  a  particular  audience,  and  a.  special 
style  of  presentation.  All  these  factors  are  pre- 
sent in  the  teaching  art.  The  aims  of  instruction, 
the  particular  facts  to  be  taught,  the  immaturity 

2 


THE  SCOPE  OF  THE  STUDY 

of  the  child  taught,  and  the  inevitable  personality 
of  the  teacher  determine  the  style  of  instruc- 
tion,   or,  to    use    our    own    "  trade   word,"    a 
method  of  teaching.    Every  teacher,  then,  has  ' 
a  style  or   method  —  conscious  or  unconscious, 
good,  bad,  or   indifferent.    Unlike  the  literary 
artist,  he  has  many  ends  to  serve  rather  than  one. 
His  functions  are  general   to  life,  and  include 
moral,  social,  and  personal  ends,  as  well  as  those 
that  are  aesthetic.  His  methods  of  communica- 
tion, too,  are  more  than  one.  He  presents  his  ex- 
periences objectively  and  graphically,  as  well  as      \ 
through  the  medium  of  written  words  and  speech. 
Always  the  teacher's  end  is  to  stimulate  growth   -~~ 
through  the  presentation  of  experiences.  When 
that  presentation  takes  a  form  and  order  differ- 
ent from  that  usual  to  adult  life  for  the  precise 
purpose  of  making  the  fact  more  readily  compre- 
hensible to  the  immature  mind  of  the  child,  then      , 
that  modification  may  be  called  a  method  of      \ 
teaching.^  Teaching  methods  are  always  special     J 
manners   of   readjusting   adult  wisdom   to  the 
special  psychological  conditions  of  a  student's 
m'md.f 

3 


\ 


TEACHING  PRIMARY  ARITHMETIC 

Distinct  Uniformities  Exist  among  its  Variations 

In  the  concrete,  methods  of  teaching  are  always 
specialized  responses  to  situations,  and  as  variable 
as  situations  are  variable.  Life  is  never  just  the 
same  at  any  point.  Yet  certain  essential  similari- 
ties exist  and  give  us  the  opportunity  to  inter- 
pret life  in  terms  of  law.  The  same  may  be  said 
of  the  teaching  life.  In  a  sense  it  never  repeats 
itself ;  yet  to  the  degree  that  the  same  end,  the 
same  subject  matter,  and  the  same  immaturity  of 
mind  recur  in  class  rooms,  teachers  will  tend  to 
use  similar  modes  of  adjustment.  In  describing 
mathematical  teaching  in  the  primary  schools,  it 
is  these  similar  modes  of  teaching  adjustment, 
these  similar  "  general  methods,"  that  we  shall 
describe  and  analyze. 

The  Methods  of  Public  Elementary  Schools  are 
Representative 

It  will  be  unnecessary  to  have  a   separate 
treatment  of  the  "general  methods  "of  mathe- 
matical teaching  for  public  schools  and  private 
schools.  Whatever  may  be  said  of  the  state-sup- 
4 


THE  SCOPE  OF  THE  STUDY 

ported  schools  will  in  general  be  true  of  private 
institutions.  It  is  true  of  elementary  schools  as 
it  is  not  of  secondary  and  higher  schools,  that 
private  institutions  hold  a  relatively  minor  place, 
as  compared  with  public  or  state  schools.  They 
are  in  a  sense  mere  adjuncts  to  the  public  school 
system,  claiming,  in  the  generality  of  cases,  no 
real  difference  in  their  ideals  and  methods  of  in- 
struction. In  the  larger  number  of  cases  they 
draw  their  courses  of  study  and  methods  from 
the  public  school  systems  of.  the  immediate  neigh- 
borhood. The  social  jtatus  of  the  parent  or  the 
personal  incapacity  of  the  child,  rather  than 
difference  of  school  methods,  is  the  cause  of  the 
special  clientage  of  the  private  primary  schools  in 
the  United  States.  Hence,  a  description  of  the 
characteristic  methods  of  the  public  schools  will 
be  representative  of  the  prevailing  modes  of  in- 
struction in  American  private  elementary  schools. 

Elementary  Mathematics  is  Mainly  Arithmetic 

Mathematical  instruction  in  the  first  six  years 
of  the  elementary  school  concerns  itself  almost 
exclusively  with  the  teaching  of  arithmetic.  A 
5 


TEACHING  PRIMARY  ARITHMETIC 

decade  or  so  ago  there  was  a  vigorous  movement 
for  the  introduction  of  algebra  and  geometry 
into  the  elementary  school.  As  a  result,  these 
subjects  made  their  appearance  in  the  seventh 
and  eighth  grades  —  seldom  in  the  first  six 
school  years.  Where  the  influence  of  this  move- 
ment penetrated  the  lower  grades  or  persisted 
in  the  higher  grades,  the  algebraic  and  geometric 
elements  involved  were  so  restricted  and  simpli- 
fied that  they  became  part  and  parcel  of  the 
subject  of  arithmetic,  rather  than  the  elemen- 
tary phases  of  two  more  advanced  subjects.  This 
is  true  of  the  simple  algebraic  equation  or  the 
measurement  of  simple  geometric  figures  where 
introduced  below  the  seventh  year. 

Elementary  Arithmetic  Emphasizes  the  Four 
Fundamental  Processes 

By  common  practice,  even  the  arithmetic 
taught  in  the  primary  grades  has  been  given  a 
restrictive  emphasis.  For  the  most  part  it  is  con- 
cerned with  the  mastery  of  the  fundamental  pro- 
cesses in  manipulating  integers  and  fractions. 
The  casual  observer,  in  reading  American  courses 
6 


THE  SCOPE  OF  THE  STUDY 

of  study,  will  note  that  in  the  lower  grades  the^ 
mathematical  subjects  taught  are  named  after  ( 
the  abstract  process  involved  rather  than  after  \ 
the  particular  business  institution  to  which  the     ) 
arithmetic   is  concretely  applied.  Thus  in   the~^ 
lower  grades  we   teach   counting,   subtraction, 
fractions,  decimals,  percentage,  etc. ;  while  in 
the  higher  grades  we  teach  interest,  stocks  and 
bonds,  commission,  insurance,  etc.  There  is  of 
course  no  hard  and  fast  line  of  demarcation ;  one 
emphasis  gradually  passes  over  into  the  other, 
an  approximate  balance  being  maintained  in  the 
intermediate  grades.  It  will  perhaps  simplify  the 
task  of  this  study  and  make  its  treatment  more 
thoroughly  representative   of  all  conditions,  if 
the  general   methods    described   be   restricted 
to  that  field  which  is  most  characteristic  of  the 
first  five  or  six  years  of  mathematical  instruction, 
namely,  to  the  teaching  of  the  fundamental  pro- 
cesses  of  manipulating  integers  and  fractions 
along  with  their  simple  applications  to  concrete 
problems.   This  discussion,  then,  will  be  limited 
to  the  period  of  school  life  in  which  the  tools  of 

arithmetic  are  acquired. 
i 

7 


TEACHING  PRIMARY  ARITHMETIC 

The  Need  for  Studying  Exceptional  Reform 
Tendencies 

While  the  aspects  of  mathematical  instruction 
here  studied  and  presented  are  selected  because 
of  their  representative  nature,  it  would  be  un- 
wise to  restrict  ourselves  to  a  statement  of  the 
commonly  accepted  procedures  of  school-room 
practice.  There  are  in  America  certain  reform 
tendencies  which  are  as  characteristic  of  condi- 
tions as  are  the  conservative  practices.  These 
modifying  forces  need  to  be  mentioned  along 
with  the  practices  that  they  alter.  Again,  there 
are  certain  scientific  effects  now  well  under 
way  to  study  the  problem  of  methods  in  teach- 
ing. While  these  have,  as  their  immediate  aim, 
the  acquisition  of  new  knowledge  rather  than  di- 
rect educational  reform,  their  ultimate  effect  will 
be  to  change  methods  of  teaching.  For  this  reason 
they  are  important,  and  have  a  proper  place  in 
this  presentation. 


II 

THE  INFLUENCE  OF  AIMS  ON  TEACHING 


.  ^^ 
Factors  Influencing  Teaching  Methods 

IT  has  been  suggested  that  all  teaching  methods 
represent  adjustments  to  several  variable  factors 
in  the  school-room  situation.  Teaching  method 
is  never,  or  should  not  be,  just  one  thing.  It  is  as 
variable  as  the  factors  that  determine  its  situa- 
tion. The  purposes  of  mathematical  instruction,--- 
the  nature  of  the  fact  to  be  taught,  the  imma- 
turity of  the  child,  the  teacher's  scholarly  equip- 
ment, his  personality,  his  attitude  toward  the 
very  idea  or  institution  of  method, — all  these  are 
influences  in  determining  the  status  of  mathe- 
matical teaching.  Some  of  them  are  so  important 
that  it  will  be  necessary  to  discuss  them  in  de- 
tail at  the  very  outset. 

The  Influence  of  a  Scientific  Aim 

The  purposes  of  mathematical  instruction  in 
the  elementary  school  must  always  be  very  in- 
9 


TEACHING  PRIMARY  ARITHMETIC 

fluential  upon  method.  It  makes  a  great  differ- 
ence whether  one  is  merely  teaching  the  elements 
of  mathematics  or  is  teaching  mathematics  as  a 
tool  for  business  life.  It  has  not  been  long  since 
the  aim  of  mathematical  teaching  was  merely 
scientific.  The  facts  taught  were  the  beginning 
of  a  science,  and  the  end  was  to  obtain  a  foun- 
dation for  more  advanced  facts  of  the  same  kind, 
which  were  dependent  upon  this  foundation.  As 
the  teacher  had  learned  his  mathematics,  so  he 
taught  the  subject.  To  a  considerable  degree,  as 
the  master's  adult  mind  classified  the  facts  of  the 
subject,  so  he  presented  it  to  the  child.  His 
methods  were  logical  rather  than  psychological. 
He  gave  the  finished  product  or  process  to  the 
child  without  special  adaptation  to  the  child's 
immaturity;  a  roundabout  method  tha£  slowly 
approximated  and  only  finally  achieved  the  full 
result  was  with  such  a  teacher  exceptional. 

Such  a  scientific  aim,  implicit  rather  than  ex- 
pressed, dominated  the  methods  of  teaching  when 
arithmetic  was  handed  over  to  the  elementary 
schools  by  the  higher  institutions  of  education. 
The  first  purpose  to  be  rooted  in  the  traditions 
10 


INFLUENCE  OF  AIMS  ON  TEACHING 

of  mathematical  teaching,  it  still  persists  with 
all  the  rigidity  of  a  conservative  force.  Teachers 
still  tend  to  teach  future  workmen  in  the  lower 
schools.as  they  themselves  were  taught  by  sci- 
entific scholars  in  the  universities.  And  high 
school  and  college  instructors  still  impose  their 
standards  upon  the  lower  schools  so  as  to  influ- 
ence their  methods  of  instruction.  As  higher  in- 
struction still  remains  largely  scientific  in  purpose 
and  method,  its  effects  reinforce  the  earliest  tra- 
dition in  the  elementary  schools.  Under  such 

an  influenceithe  worth  of  a  mathematical  fact  is--' 

vt'jr 

measured  by  its  place  in  a  logical  scheme,  rather} 
than  by  its  significance  and  recurrence  in  every-' 
day  life.\  The  mathematician  may  need  to  know 
all  abour  the  names  of  the  places  in  notation  and 
numeration  ;  the  layman  cares  only  about  the  ac- 
curate reading  and  writing  of  numbers,  and  not 
at  all  about  the  verbal  title  of  "units  of  thou- 
sands "  place.  Again  the  rational  needs  of  a  stu- 
dent of  mathematics  may  require  an  understand- 
ing of  the  reasons  why  we  "carry"  in  column 
addition,  but  the  effective  everyday  use  demands 
an  accurate  habit  of  "carrying"  rather  than  an 
ii 


TEACHING  PRIMARY  ARITHMETIC 

accurate  explanation.  Yet  just  such  methods 
persist  in  our  schools  because  of  the  domination 
of  a  scientific  treatment  of  the  subject. 

The  Influence  of  the  Aim  of  Formal  Discipline 

The  remoteness  of  such  mathematical  teach- 
ing from  the  needs  of  common  life  constantly 
threatens  the  loyalty  and  support  of  the  public. 
Some  defense  becomes  necessary  on  other  than 
scientific  grounds.  Such  a  sanction  could  not  be 
found  in  utilitarianism,  for  the  waste  was  evi- 
dent. It  remained  for  a  psychological  theory  to 
sketch  a  defense  upon  "disciplinary"  grounds. 
The  doctrine  of  "formal  discipline"  says  that 
such  mathematical  teaching  trains  the  powers  of 
the  mind  so  that  any  mastery  gained  in  mathe- 
matics is  a  mastery  operating  in  full  elsewhere, 
regardless  of  the  remoteness  of  the  new  situa- 
tions from  those  in  connection  with  which  the 
power  or  ability  was  originally  acquired^)  The  facts 
and  processes  mastered  may  not  be  those  most 
needed  in  daily  life,  but  they  are  good  for  every 
nan  inasmuch  as  they  train  his  mind.)  Such  was 
.he  dictum  of  the  doctrine  of  "  f  ormaTdiscipline." 

12 


INFLUENCE  OF  AIMS  ON  TEACHING 

The  effect  of  such  doctrine  is  to/defend  and 
perpetuate  every  obsolete,  unimportant,  and 
wasteful  practice)in  the  teaching  of  mathematics.^ 
No  matter  that  partnership  as  taught  in  the 
schools  had  its  original  sanction  in  its  close  cor- 
respondence to  the  reality  of  business  practice ; 
no  matter  that  the  old  sanction  has  passed ; 
teach  it  now  for  its  ability  to  discipline  the* mind ! 
No  matter  that  "  life  insurance "  touches  more 
men  than  "cube  root"  ;  the  latter  should  be  kept 
because  of  its  power  to  train  the  mind.  In  life, 
where  "approximation"  of  amounts  suffices,  the 
teacher  demands  absolute  accuracy,  and  the  eth- 
ical worth  of  such  precise  truth  is  the  high  law 
for  its  defense.  In  life,  "the  butcher,  the  baker, 
and  the  candlestick  maker  "  figure  out  the  total 
of  a  bill  mainly  "in  their  heads,",  with  a  few  ac- 
cessory pencil  scribbles  upon  paper;  the  teacher 
finds  sanction  in  aesthetics  for  requiring  a  com- 
plete statement  written  or  re- written  in  exquisite 
form.  Regardless  of  the  truth  that  is  concealed 
in  the  doctrine  of  "  formal  discipline,"  it  must  be 
confessed  by  those  who  know  the  history  of 
teaching  method  in  the  United  States  that  it  is 
13 


TEACHING  PRIMARY  ARITHMETIC 

the  main  defense  of  conservatism  and  the  larg- 
est cause  of  waste  in  teaching  methods. 

The  Shift  in  Emphasis  from  Academic  to 
Social  Aims 

Such  has  been  the  ground  upon  which  recent 
educational  reform  has  operated.  Slowly  the 
\  older  scientific  and  disciplinary  aims  of  instruc- 
\  tion  have  given  way  to  the  newer  purposes  of 
business  utility  and  social  insight.  In  that  step 
a  large  transition  has  been  covered.  Before,  the 
school  measured  the  worth  of  its  work  by  stand- 
ards internal  to  educational  institutions.  The 
schoolmaster  and  the  scholar,  rather  than  the 
man  on  the  street,  had  formulated  the  scientific 
classifications  of  mathematics  and  expounded 
the  doctrine  of  "formal  discipline."  Thereafter, 
the  measure  of  efficient  school  instruction  was 
determined  by  standards  external  to  the  school, 
the  product  of  conditions  outside  of  school  life. 
\J3usiness  need  and  social  situation  determine 
whether  a  fact  or  a  process  is  worth  comprehend- 
ing, and  whether  the  method  of  instruction  has 
been  effective?\ 

14 


INFLUENGE  OF  AIMS  ON  TEACHING 

Business  Utility  as  an  End 

The  utilitarianism  that  first  attacked  the  older 
course  of  study  and  its  methods  was  the  utility  of 
the  business  world.  The  arithmetic  of  business  life- 
became  the  standard.  The  practices  of  the  market 
determined  what  matter,  skill,  and  accuracy 
should  be  demanded  of  the  elementary  school 
pupil.  Recently  it  became  the  habit  to  call  upon 
the  business  man  to  give  his  opinion  as  to  what 
constitutes  good  arithmetical  training;  and  no 
criticism  was  so  feared  as  that  of  the  business 
leader  who  said  that  the  boys  that  came  to  him 
were  incompetent.  Committees  on  courses  of 
study  have  even  investigated  the  relative  fre- 
quency and  importance  of  specific  arithmetical 
processes  in  the  business  world  with  the  idea  of 
utilizing  the  results  as  a  basis  for  changes  in  the 
mathematical  curriculum. 

This  aim  of  business  utility,  coming  at  a  time 
when  the  elementary  school  course  was  felt  to 
be  overcrowded,  met  with  a  ready  reception.  It 
operated  for  the  time  being  as  the  standard  by 
which  materials  and  methods  in  arithmetic  were 
15 


TEACHING  PRIMARY  ARITHMETIC 

to  be  eliminated,  if  not  actually  selected.  Ma- 
terials not  general  to  the  business  world,  such 
as  the  table  of  Troy  weight,  were  therefore 
eliminated.  Processes  of  computing  interest  in- 
frequently used  were  supplemented  by  more 
widespread  and  up-to-date  methods.  More  doing 
and  less  explaining  characterized  the  instruction 
in  adding  columns  of  figures,  and  such  manipu- 
lation mimicked  the  exact  conditions  of  its  use 
in  the  world  at  large.  If  strings  of  figures  are 
usually  added  in  vertical  columns  in  the  busi- 
ness world,  then  they  should  be  taught  in  ver- 
tical columns  more  nearly  exclusively  than  before. 
The  obsolete  and  the  relatively  infrequent,  the 
over-complex  and  the  wasteful  processes  of  the 
old  arithmetic  tended  to  disappear.  More  than 
any  other  influence,  this  aim  of  business  utility 
has  combated  the  over-conservative  influence  of 
scientific  and  disciplinary  aims  which  dominated 
previous  decades.  The  newer  methods  of  teach- 
ing have  kept  the  best  of  the  old  movements. 
The  work  is  still  scientific  in  that  it  is  accurate; 
it  is  still  disciplinary  in  that  it  trains ;  but  the 
truth  and  the  training  which  are  given  are 
16 


INFLUENCE  OF  AIMS  ON  TEACHING 

selected  by  and  associated  with  actual  business 
situations  common  to  every-day  life. 

Broad  Social  Utilitarianism  as  a  Standard 

There  is  evidence  in  the  present  thought  of 
teachers  that  a  utility  broader  than  that  of  the 
business  world  is  beginning  to  obtain  in  the 
schools.  Everywhere  in  these  days  the  Ameri- 
can teacher  and  the  educational  writer  speak  of 
the  social  aims  of  education.  More  than  ever  be- 
fore, the  social  consciousness  of  the  teacher  en- 
larges. This  general  increase  in  the  social  con- 
sciousness of  the  teacher  is  reflected  in  mathe- 
matical instruction. 

In  spite  of  an  increasing  movement  toward 
specific  vocational  training,  —  as  seen  in  the  in- 
dustrial education  movement,  for  example, — 
there  has  been  a  reactionary  defense  of  the  ele- 
mentary schools  as  an  institution  for  very  gen- 
eral training  in  the  things  that  arc-socially  fun- 
damental and  common.  This  movement  toward 
the  preservation  of  the  elementary  school  as  a 
place  for  giving  a  broadly  socialized  and  modern 
culture  not  only  is  checking  the  inroads  of  a 
17 


TEACHING  PRIMARY  ARITHMETIC 

narrow  vocational  education,  but  is  broadening 
the  conception  of  the  older  studies,  of  which 
arithmetic  is  but  one.  Arithmetic  is  not  a  sub- 
ject in  which  only  the  skills  of  calculation  are 
cultivated ;  it  is  one  that  contributes  social  in- 
sight, just  as  history  and  geography  do. 

The  influence  of  the  social  aim  of  instruction 
upon  mathematical  instruction  is  subtle  but  ob- 
vious. The  business  man's  opinion  with  refer- 
ence to  arithmetical  instruction  is  not  always 
taken  as  gospel.  There  are  other  standards. 
"  Why,"  says  the  schoolmaster,  "  should  I  train 
people  for  your  special  needs,  any  more  than  for 
the  demands  of  other  trades  that  men  ply?  To 
be  sure,  our  graduates  do  not  fit  perfectly  into 
your  shop  at  once.  But  that  precise  and  local 
adjustment  is  the  work  of  the  business  course 
or  of  shop  apprenticeship.  My  function  is  to 
train  men  for  the  situations  common  to  all  men 
and  special  to  no  class.  The  elementary  school 
is  a  school  for  general  culture  or  social  apprecia- 
tion, not  a  business  college  or  a  trade  school." 
The  sociologist  usurps  the  place  of  the  business 
man  as  the  school's  proper  critic. 
18 


INFLUENCE  OF  AIMS  ON  TEACHING 

Some  Concrete  Effects  of  the  Change  in  Aim 

The  immediate  effect  is  that  arithmetical  ap- 
plications find  a  larger  place  in  teaching.  A 
saner  relation  is  established  between  abstract 
examples  and  concrete  problems.  And  the  prob- 
lems, in  increasing  extent,  are  real  problems, 
typical  of  life,  if  not  actual.  No  more  does  arith- 
metic, in  the  best  schools,  confine  itself  to  fig- 
ures alone.  Figures  are  applied  in  concrete 
problems.  There  may  be  days  of  teaching  when 
not  a  figure  is  used  during  the  arithmetic  period. 
The  social  setting,  the  business  institution,  which 
calls  for  the  calculation,  is  studied  as  carefully 
as  the  process  of  calculation.  The  students  are 
given  a  knowledge  of  banking  as  well  as  skill  in 
the  computation  of  interest.  They  may  even 
visit  a  bank,  a  factory,  a  shop,  as  the  case  may 
require.  Instead  of  having  fifteen  problems  that 
.deal  with  fifteen  different  subjects  all  more  or 
less  remote  from  one  another,  as  was  almost  uni- 
versally the  case  with  older  text-books  and  teach- 
ing methods,  the  class  hour  may  be  given  over  to 
fifteen  problems  related  to  one  situation,  such 
19 


TEACHING  PRIMARY  ARITHMETIC 

as  might  develop  in  the  business  of  a  bakery 
shop  or  an  apartment  house.  Thus  arithmetic 
gradually  gains  social  setting  and  unity. 

To-day  teaching  methods  in  arithmetic  are  in 
a  state  of  transition  :  old  and  new  purposes  min- 
gle with  unequal  force  and  give  us  a  mixed  pro- 
cess of  instructing.  Old  materials  and  methods 
still  persist,  for  logical  and  disciplinary  ideals 
still  hold ;  but  the  newer  regimen  ushered  in  by 
the  demands  of  business  utility  and  social  under- 
standing gains  ground.  The  obsolete,  the  untrue, 
the  wasteful  methods  pass  from  arithmetic  teach- 
ing ;  and  the  pressing,  tfiodern,  and  useful  activi- 
ties and  understandings  enter.  Arithmetic  is  less 
abstract  and  formal  as  a  subject  than  it  was ;  it 
has  become  increasingly  vital  and  concrete  with 
real  interests,  insights,  and  situations.  The  grind 
of  sheer  mechanical  drill  decreases  in  teaching ; 
and  a  reasoned  understanding  of  relations,  in 
some  degree  at  least,  is  substituted.  Artificial  mo- 
tives and  incentives  are  less  frequently  used  to 
get  work  done,  while  the  quantitative  needs  of  the 
child's  life  and  the  intrinsic  interest  of  children 
in  the  institutional  occupations  of  their  elders  pro- 
vide a  more  vital  motive  for  the  use  of  arithmetic. 


Ill 

THE  EFFECT  OF  THE  CHANGING  STATUS  OF 
TEACHING  METHOD 

Method  as  a  Psychological  Adjustment  to 
the  Child 

AT  the  very  beginning,  it  was  suggested  that 
many  factors  enter  into  the  nature  of  our  teach- 
ing methods.  There  was  occasion  to  show  the 
effect  of  varying  aims  on  the  spirit  and  manner 
of  instruction,  for  the  end  in  view  inevitably  in- 
fluences any  presentation  of  facts,  in  school  or 
out.  The  most  significant  factor,  however,  in 
teaching  method  is  the  attempt  to  adjus^  meth- 
ods of  presentation  to  the  psychological  con- 
ditions of  childhood.  Teaching  method  in  the 
school  is  primarily  a  mode  of  presentation  de- 
signed to  stimulate  the  energies  of  children.  As 
long  as  the  teacher  was  the  most  active  person 
in  the  classroom,  method  as  such  was  not  impor- 
tant in  pedagogical  theory.  The  focusing  of  at- 
tention on  the  child  as  an  active  human  factor  to 

21 


TEACHING  PRIMARY  ARITHMETIC 

be  given  careful  consideration  is  responsible  for 
the  extended  development  of  teaching  technique. 
The  growing  importance  of  "method"  in  educa- 
tional theory  marks  a  growth  in  the  teacher's 
consciousness  of  psychological  factors,  precisely 
as  the  appearance  of  the  newer  aims  in  teaching 
has  marked  an  increased  regard  for  social  factors. 

The  Effect  of  an  Increased  Reverence  for 
Childhood 

Two  important  movements  have  been  respon- 
sible for  the  development  of  a  psychological  con- 
sciousness of  the  pupil  as  a  dominating  factor 
in  teaching  method.  One  is  humanitarian ;  the 
other,  scientific. 

There  has  been  a  steady  growth  in  reverence 
and  sympathy  for  childhood.  As  yet  it  has 
scarcely  expressed  itself  with  fullness.  Its  pre- 
sence is  revealed  by  the  widespread  enactment  of 
laws  designed  to  guarantee  the  rights  of  child- 
hood —  laws  against  child  labor  and  in  favor  of 
compulsory  education.  The  growth  of  special 
courts  for  juvenile  offenders,  the  development  of 
playgrounds,  and  the  decreased  brutality  of  dis- 
22 


CHANGING  STATUS  OF  METHOD 

cipline  at  home  and  school,  are  other  symptoms 
of  the  public  attitude  toward  childhood.  The 
wide  acceptance  of  the  "  doctrine  of  interest " 
in  teaching ;  the  enrichment  of  the  curriculum  ; 
specialized  schools  for  truants  and  defectives; 
individual  instruction,  —  these  are  the  school- 
master's recognition  of  the  modern  attitude  to- 
ward childhood.  Under  such  conditions  teach- 
ing becomes  less  and  less  a  ruthless  external 
imposition  of  adult  views,  and  more  a  means 
of  sympathetic  ministry  to  those  inner  needs  of 
child  life  which  make  for  desirable  qualities  of 
character.  While  it  is  true  that  teaching  method 
is  a  condescension  to  childhood,  it  is  a  socially 
profitable  condescension  in  that  it  is  a  guarantee 
of  more  effective  and  enduring  mastery  of  the 
life  that  is  revealed  at  school.  Since  the  child's 
acquisition  tends  the  more  to  be  part  and  parcel 
of  his  own  life  under  such  sympathetic  teaching, 
the  products  of  such  instruction  are  enduring. 

The  Reconstruction  of  Method  through  Psychology 

Such    a    humanitarian    movement    naturally 
called  for  knowledge  of  the  child  —  the  wisdom 
23 


TEACHING  PRIMARY  ARITHMETIC 

of  common  sense  soon  exhausts  itself,  and  more 
scientific  data  are  demanded.  Thus  the  "  child 
study  movement"  came  into  existence.  The 
movement  was  in  some  degree  disappointing,  for 
frequently  it  busied  itself  in  cataloguing  the  ob- 
vious rather  than  in  classifying  new  and  hitherto 
unexplained  data.  But  one  thing  it  did  :  it  focused 
attention  upon  the  child  as  the  crucial  factor 
in  education,  the  prime  conditioning  force  in  all 
methods  of  instruction.  Since  then,  a  saner  psy- 
chological foundation  has  been  laid  for  educa- 
tional procedure,  one  which  is  criticising  and 
reconstructing  teaching  method  at  every  turn. 
Hitherto,  teaching  methods  had  been  improved 
fitfully  through  a  crude  empiricism.  As  the 
ablest  teachers  became  dissatisfied  with  their 
teaching  and  dared  to  vary  their  methods,  they 
selected  the  successful  experiments,  and  other 
teachers  willingly  adopted  the  methods  that 
seemed  better  than  their  own.  Now  a  body  of 
general  psychological  knowledge,  rich  in  its 
criticism  of  old  methods  and  in  its  suggestion 
of  new  means  of  procedure,  gives  a  scientific 
basis  to  teaching  method.  Where  additional  psy- 
24 


CHANGING  STATUS  OF  METHOD 

chological  knowledge  is  needed,  the  educational 
psychologists  seek  it  through  special  investiga- 
tions. And  where  the  counter  claims  of  com- 
peting methods  defy  ordinary  psychological 
analysis  and  investigation,  judgment  is  sought 
through  an  experimental  pedagogy  which  sub- 
mits teaching  processes  to  comparative  tests 
under  normal  classroom  conditions. 

The  Increased  Professional  Respectability  of 
Conscious  Method 

Increased  sympathy  with  childhood  and  in- 
creased scientific  knowledge  of  human  nature 
together  give  teaching  method  a  new  j  ustification. 
The  result  is  that  the  era  of  complete  depend- 
ence upon  teaching  genius  and  mere  common 
sense  in  methods  of  instruction  has  passed  out 
of  the  American  elementary  school.  We  are  now 
in  a  period  where  a  specific  professional  tech- 
nique in  teaching  is  demanded,  a  technique  partly 
developed  out  of  crude  personal  and  professional 
experience,  and  partly  founded  upon  scientific 
criticism  and  experiment.  A  new  humanitarian 
and  scientific  attitude  toward  the  mental  life  of 
25 


TEACHING  PRIMARY  ARITHMETIC 

children  elevates  teaching  method  to  a  position 
it  has  never  before  enjoyed. 

The  public  elementary-school  teacher  is  con- 
servative indeed  who  will  deny  that  there  is  any- 
thing worthy  in  the  notion  of  "  method."  As  a 
class,  teachers  have  faith  in  the  special  profes- 
sional technique  which  is  included  under  the 
term.  They  are  critical  of  the  many  abuses  which 
have  been  committed  in  the  name  of  method. 
Method  cannot  be  a  substitute  for  scholarship. 
It  cannot  be  a  "cut  and  dried"  procedure  indis- 
criminately or  uniformly  applied  to  class-room 
instruction.  Like  every  other  technical  means, 
teach  ing  method  is  subject  to  its  own  limitations 
and  strengths,  a  fact  which  the  average  teacher 
recognizes. 

The  Prevalence  of  Methods  Emphasizing  a 

Single  Idea 

*-. 

In  spite  of  the^act  that  the  majority  of  ele- 
mentary teachers  Keep  reasonably  sane  on  the 
problem  of  method  in  teaching,  it  must  be  ad- 
mitted that  a  considerable  proportion  of  teachers 
are  inclined  to  be  attracted  by  systems  of  method 
26 


CHANGING  STATUS  OF  METHOD 

that  greatly  over-emphasize  a  single  element  of 
procedure.  The  hold  which  the  "  Grube  method," 
with  its  unnatural  logical  thoroughness  and 
progression,  gained  in  this  country  two  or 
three  decades  ago  is  scarcely  explicable  to-day. 
Scarcely  less  baffling  is  the  very  large  appeal 
made  by  a  series  of  textbooks  which  lays  the 
stress  upon  the  acquisition  of  arithmetic  through 
the  idea  of  ratio  and  by  means  of  measuring. 
Manual  work  as  the  source  of  arithmetical  ex- 
periences is  another  special  emphasis,  which,  like 
the  others,  has  had  its  enthusiastic  adherents. 
Again  it  is  "arithmetic  without  a  pencil"  or 
some  other  over-extension  of  a  legitimate  local 
method  into  a  "panacea  "or  "cure-all,"  which  con- 
fronts us.  The  promulgation  and  acceptance  of 
such  unversatile  and  one-sided  systems  of  teach- 
ing method  are  indicative  of  two  defects  in  the 
professional  equipment  of  teachers  :  (i)  the  lack 
of  a  clear,  scientific  notion  as  to  the  nature  and 
function  of  teaching  method,  and  (2)  the  lack  of 
psychological  insight  into  the  varied  nature  of 
class-room  situations.  Untrained  teachers  we  still 
have  among  us,  and  others,  too,  to  whom  a  little 
27 


TEACHING  PRIMARY  ARITHMETIC 

knowledge  is  a  dangerous  thing.  These  are  fre- 
quently carried  away  by  the  enthusiastic  appeals 
of  the  reformer  with  a  system  far  too  simple  to 
meet  the  complex  needs  of  human  nature.  Our 
experiences  seem  to  have  sobered  us  somewhat, 
the  increase  of  supervision  has  made  responsible 
officers  cautious,  and  increased  professional  in- 
telligence has  put  a  wholesome  damper  upon 
na'fve  and  futile  proposals  to  make  teaching 
easy. 

The  Tendency  toward  Over-Uniformity 
in  Method 

A  more  serious  evil  than  that  just  mentioned 
is  the  tendency  of  the  supervising  staff  to  over- 
prescribe  specific  methods  for  class-room  teach- 
ers. Recently  there  has  developed,  more  par- 
ticularly in  large  city  systems,  a  tendency  to 
demand  a  uniform  mode  of  teaching  the  same 
school  subject  throughout  the  city.  The  prime 
causes  of  this  tendency  are  to  be  found  (i)  in  the 
specialization  of  grade  teaching,  and  the  conse- 
quent interdependence  of  one  teacher  on  another; 
and  (2)  in  the  mobility  of  the  school  population, 
28 


CHANGING  STATUS  OF  METHOD 

which  involves  considerable  lost  energy  if  teachers 
do  not  operate  along  similar  lines. 

The  result  of  such  imposed  uniformity  is  a 
reduction  of  spontaneity  in  teaching.  The  pro- 
cess of  instruction  proceeds  in  a  more  or  less 
mechanical  fashion,  the  teacher  working  for  bulk 
results  by  a  persistent  and  general  application  of 
the  methods  laid  down.  That  teaching  which  at 
every  moment  tends  to  adjust  itself  skillfully  to 
the  changes  of  human  doubt  and  interest,  diffi- 
culty and  success,  discouragement  and  insight, 
now  taking  care  of  a  whole  group  at  once,  now 
aiding  an  individual  straggler,  now  resolutely 
following  a  prescribed  lead,  now  pursuing  a  line 
of  least  resistance  previously  unsuspected,  can- 
not thrive  under  such  conditions.  The  demand 
for  an  excessive  uniformity  stifles  teaching  as 
a  fine  art,  and  makes  of  it  a  mechanical  busi- 
ness ;  only  those  activities  that  fit  the  machine 
can  go  on.  Thus  it  happens  that  we  memorize, 
cram,  drill,  and  review ;  and  soon  the  subtler  pro- 
cesses of  thinking  and  evaluating,  which  are 
the  best  fruit  of  education,  cease  to  exist. 


TEACHING  PRIMARY  ARITHMETIC 

Method  as  a  Series  of  Varied,  Particular 
Adjustments 

Fortunately  the  one-method  system  of  teach- 
ing will  soon  belong  to  the  past ;  and  fortunately, 
too,  the  imposition  of  uniform  methods  is  begin- 
ning to  lose  ground,  even  in  our  cities.  For  the 
most  part,  the  common  sense  of  teachers  and 
the  positive  statements  of  our  better  theorists 
keep  teaching  methods  in  a  sane  and  useful 
status.  Teaching  methods  should  be  as  infinitely 
variable  as  the  conditions  calling  for  their  use  are 
endlessly  changeable.  Not  one  method  but  many 
are  necessary,  for  methods  are  supplementary 
rather  than  competitive.  No  one  method  should 
be  used  with  a  pre-established  rigidity ;  each  must 
be  flexible  in  its  uses,  so  as  to  accomplish  the 
varied  work  to  be  done.  The  teacher,  with  his 
everyday  contact  with  the  problems  of  child- 
hood, is  the  best  interpreter  of  conditions  and 
the  best  chooser  of  the  tools  of  instruction. 
The  supervisor  may  criticize,  suggest,  and  ad- 
vise ;  he  may  call  attention  to  fundamental  prin- 
ciples involved ;  but  the  teacher  himself  must 


CHANGING  STATUS  OF  METHOD 

finally  choose  his  own  methods.  He  is  the  only 
one  who  can  know  conditions  well  enough  to  ad- 
just teaching  methods  to  the  needs  of  his  own 
children. 

Arithmetic  teaching  has  suffered  from  false 
uses  of  teaching  method.  In  this  respect  it  has 
shared  the  common  professional  lot.  But  in 
addition  it  has  had  special  difficulties  and  adven- 
tures of  its  own.  We  have  now  to  note  those 
special  phases  of  teaching  method  which  are 
peculiar  and  local  to  mathematical  instruction. 

J  f       ' 


IV 

METHOD  AS   AFFECTED  BY  THE  DISTRIBIK 
TION  AND  ARRANGEMENT  OF  ARITH- 
METICAL WORK 

The  Tendency  toward  Shortening  the  Time 
Distribution 

SEVERAL  decades  ago  arithmetic,  as  a  formal 
subject,  was  begun  in  the  first  school  year  and 
continued  throughout  the  grades  to  the  last 
school  year.  This  is  no  longer  a  characteristic 
condition,  much  less  a  uniform  one.  There  have 
been  forces  operating  to  complete  the  subject  of 
arithmetic  prior  to  the  eighth  year,  and  to  delay 
its  first  systematic  presentation  in  the  primary 
grades  for  a  period  varying  from  six  months  to 
two  years.  The  report  of  the  "Committee  of 
Fifteen  "  of  the  National  Education  Association 
summarizes  the  tendency  existing  in  1895  when 
it  states  that,  "with  the  right  methods,  and  a 
wise  use  of  time  in  preparing  the  arithmetic 
lesson  in  and  out  of  school,  five  years  are  surfi- 
32 


DISTRIBUTION  AND  ARRANGEMENT 

cient  for  the  study  of  mere  arithmetic  — the  five 
years  beginning  with  the  second  school  year  and 
ending  with  the  close  of  the  sixth  year. 

The  Attempt  to  Eliminate  Waste 

The  attempt  to  shorten  the  period  of  formal 
instruction  in  arithmetic  has  had  its  effects  upon 
the  methods  of  teaching  as  well  as  upon  the  ar- 
rangement of  the  course  of  study.  The  presence 
of  a  large  number  of  children  who  leave  school 
by  the  seventh  year,  the  example  of  a  varied 
European  practice,  the  overcrowded  curriculum, 
— all  these  have  combined  to  suggest  a  short- 
ened treatment  of  arithmetic.  Hence  economy, 
through  the  elimination  of  obsolete  and  unim- 
portant topics  in  the  course  of  study  and  through 
better  methods  of  instruction,  has  become  a 
pressing  matter.  Its  influence  on  method  is  ob- 
vious. 

It  has  focused  attention  upon  "teaching 
method"  and  given  it  an  increasing  importance 
in  the  eyes  of  mathematics  teachers.  Specifically, 
it  has  tended  to  reduce  the  amount  of  objective 
work,  to  eliminate  the  explanation  or  rationaliza- 
33 


TEACHING  PRIMARY  ARITHMETIC 

tion  of  processes  which  in  life  are  done  auto- 
matically; it  has  made  teachers  satisfied  with 
teaching  one  manner  of  solution  where,  before, 
two  or  three  were  given  ;  it  has  laid  the  emphasis 
upon  utilizing  old  knowledge  in  new  places,  rather 
than  on  acquiring  new  means. 

Delay  in  Beginning  Formal  Arithmetic 
Teaching 

The  tendency  toward  delay  in  beginning  for- 
mal arithmetic  instruction  is  to  be  explained  in 
terms  of  several  causes.  Under  a  regimen  where 
complicated  and  obsolete  problems,  difficult 
of  comprehension,  were  common  in  elementary 
school  tests,  it  was  natural  that  teachers  should 
believe  that  arithmetic  is  too  difficult  a  subject 
for  young  children  and  that  better  results  could 
be  obtained  if  the  subject  were  not  commenced 
till  the  children  were  more  mature. 

This  belief  persists  even  after  the  curriculum 
is  purged  of  all  obsolete  and  over-complex  ma- 
terials, and  has  become  a  modern  course  of 
study  with  materials  well  within  the  compre- 
hension and  interest  of  primary  children. 
34 


DISTRIBUTION  AND  ARRANGEMENT 

The  Incidental  Method  of  Teaching 

The  by-product  of  this  belief  is  that  any  arith- 
metic taught  during  these  first  few  years  should 
be  taught  "incidentally,"  as  a  chance  accompani- 
ment of  other  studies.  Only  after  one  or  two 
years  of  incidental  work  should  the  formal  arith- 
metic instruction  be  given.  This  "  incidental " 
method  of  teaching  beginners  is  difficult  to  es- 
timate. It  has  been  so  variously  treated  that  a 
comparative  measure  of  its  worth  is  difficult  to 
obtain.  The  contention  that  children  who  are 
taught  incidentally  for  two  years  and  systemati- 
cally for  two  years  more  have  at  the  end  of 
four  years  of  school  life  as  good  a  command  of 
arithmetic  as  those  who  have  had  a  systematic 
course  through  four  school  years,  is  difficult 
to  substantiate  or  deny  on  scientific  grounds. 
Sometimes  "incidental"  teaching  required  by 
the  course  of  study  becomes  "  systematic "  in 
the  hands  of  the  teacher.  Sometimes  the  two 
years  of  "  systematic  "  teaching  which  follow  the 
incidental  teaching  mean  far  more  than  two 
years,  since  the  teachers,  in  order  to  catch  up, 
35 


TEACHING  PRIMARY  ARITHMETIC 

give  more  time  and  emphasis  to  the  subject  than 
the  relative  time-allotment  of  any  general  sched- 
ule would  seem  to  warrant.  Such  have  been  the 
facts  frequently  revealed  by  a  class-room  inspec- 
tion that  penetrates  beyond  the  course  of  study, 
the  time  schedule,  and  regulations  of  the  school 
board. 

Reactions  against  the  Plan  of  Incidental  Teaching 
In  the  lack  of  specific  comparative  measures  of 
the  worth  of  such  methods  of  instruction,  there 
is  a  growing  conviction  (i)  that  beginning  school 
children  are  mature  enough  for  the  systematic 
study  of  all  the  arithmetic  that  the  modern 
course  of  study  would  assign  to  these  grades ;  (2) 
that,  considering  the  quantity  and  quality  of  their 
experiences,  they  can  think  or  reason  quite  as 
well  as  memorize ;  and  (3)  that  what  the  school  re- 
quires of  the  child  can  be  better  done  in  a  re- 
sponsible, systematic  manner  than  by  any  hap- 
hazard system  of  "  incidental  "  instruction. 

These  reactionary  attitudes  by  no  means  imply 
a  return  to  "  systematic  "  teaching  of  arithmetic 
in  the  first  two  school  years,  nor  to  such  formal 
36 


DISTRIBUTION  AND  ARRANGEMENT 

methods  as  were  previously  employed.  Other 
grounds  forbid.  The  crude,  uninteresting  memori- 
ter  methods  of  the  past  have  gone  for  good.  Ob- 
jective work,  plays,  games,  manual  activities  make 
arithmetical  study  easier  and  more  efficient.  In- 
deed, these  newer  methods  have  been  a  large  fac- 
tor in  convincing  teachers  that  children  have  the 
ability  to  master  the  first  steps  in  arithmetic 
during  the  first  two  years.  Regardless  of  this 
change  in  prof  essional  belief,  it  is  a  fairly  general 
opinion  that  arithmetic  should  not  be  thrown 
upon  the  school-beginner  along  with  the  other 
heavy  burden  of  learning  to  read.  Learning  the 
mechanics  of  reading  is  quite  the  most  important 
part  of  the  first  school  year,  and  the  addition  of 
the  difficulties  of  another  language  —  for  such 
number  is  —  would  overburden  and  distract  the 
child.  Hence  a  common-sense  distribution  of 
burdens  and  tasks,  regardless  of  questions  of 
child  maturity,  would  delay  the  formal  and  sys- 
tematic study  of  arithmetic  a  half  or  whole  school 
year,  little  reliance  being  placed  upon  previous 
"  incidental "  acquisitions. 


37 


TEACHING  PRIMARY  ARITHMETIC 

Logical  and  Psychological  Types  of  Arrangement 

There  are  other  problems  of  method  less  con- 
cerned with  the  time  for  beginning  the  study,  or 
with  the  span  of  school  life  to  be  given  to  it. 
These  deal  with  the  arrangement  of  sub-topics 
within  the  course  of  study,  or  with  the  manner 
of  progression  from  one  aspect  of  arithmetical 
experience  to  another.  I  refer  now  to  the  vari- 
ous methods  of  planning  the  work  in  arithmetic 
from  grade  to  grade,  of  which  the  "Grube 
method  "  and  the  "  spiral "  methods  are  types. 

The  methods  that  have  been  employed  in  the 
United  States  for  the  arrangement  or  ordering 
of  topics  within  the  course  of  study  have  varied 
considerably  from  time  to  time,  but  all  these 
variations  may  be  grouped  around  two  types: 
(l)  The  "logical"  types  of  arrangement,  and  (2) 
the  "psychological"  types  of  arrangement.  If 
the  course  of  study  proceeds  primarily  by  units 
that  are  characteristic  of  the  mathematics  of  a 
mature  adult  mind,  the  type  may  be  said  to  be 
"logical."  If  the  'course  of  study  proceeds  pri- 
marily by  units  that  are  characteristic  of  the 
38 


DISTRIBUTION  AND  ARRANGEMENT 

manner  in  which  an  immature  child's  mind  ap- 
proaches the  subject,  then  the  type  may  be  said 
to  be  "psychological."  The  dominant  arrange- 
ments have  been  "  logical "  up  to  within  the  past 
two  decades.  The  older  text-books  taught  "  nota- 
tion and  numeration  "rather  thoroughly,  then  pro- 
ceeded to  a  fairly  adequate  mastery  of  "addi- 
tion," then  to  "  subtraction,"  and  so  on.  Such 
an  arrangement  is  distinctly  "logical."  So  also 
was  the  later  "  Grube  method,"  which  progressed 
by  numbers  rather  than  by  processes. 

The  courses  of  study  which  have  been  most 
familiar  to  us  in  the  past  decade  have  used  the 
"concentric  circle"  or  "spiral"  methods  of  ar- 
ranging the  sub-topics  of  arithmetic.  These  ar- 
rangements are  "psychological"  in  type.  They 
are  attempts  to  give  a  systematic  order  of  mas- 
tery which  shall  approximate  the  child's  order  of 
need  in  knowing.  Here  the  first  mathematical 
facts  and  skills  taught  are  those  that  the  child 
first  requires,  regardless  as  to  whether  they  em- 
ploy integers  or  fractions,  additions  or  divisions. 
A  little  later,  he  deals  with  the  same  subjects 
and  the  same  numbers  in  more  complicated 
39 


TEACHING  PRIMARY  ARITHMETIC 

manipulation  and  in  more  extended  application. 
The  field  is  re-covered,  as  it  were,  by  ever  widen- 
ing circles  or  by  an  enlarged  swing  of  the  "  spiral " 
progression. 

Estimates  of  Worth 

The  older  "logical"  plans  are  thorough  and 
definite  in  their  demands ;  the  teacher  always 
knows  just  what  he  is  about.  But  such  a  system 
of  procedure  is  unnatural  and  remote  from  the 
child  ;  it  lacks  appeal  and  motive.  The  child  pur- 
sues the  subject  as  a  task  laid  down  for  him,  not 
as  an  answer  to  his  own  curiosities  or  necessities. 
The  newer  psychological  plans  meet  the  different 
levels  of  child-maturity  effectively;  they  are 
nearer  the  natural  order  of  acquiring  knowledge. 
The  difficulty  with  all  psychological  arrange- 
ments is  that  the  teacher  cannot  readily  re- 
member what  the  child  has  and  has  not  been. 
The  supervisor,  too,  finds  it  hard  to  locate  re- 
sponsibility for  the  teaching  of  definite  arith- 
metic sub-topics.  As  orders  of  teaching  they  are 
psychologically  natural  but  administratively  in- 
effective. 

40 


DISTRIBUTION  AND  ARRANGEMENT 

The  Present  Mixed Metho&of  Procedure 

The  result  is  that,  to-day,  the  two  types  of 
arrangement  are  modifying  each  other  and  giv- 
ing a  mixed  method,  partly  "  logical "  and  partly 
"psychological."  That  line,  of  least  resistance  in 
which  the  children  study  arithmetical  facts  and 
processes  with  greatest  success  is  modified  by 
definite  demands  that  topics  —  e.g.,  addition  — 
be  mastered  thoroughly  "  then  and  there."  The 
method  is  partly  "  topical "  and  partly  "  spiral." 
The  child  in  the  second  grade  may  have  a  little 
of  all  the  fundamental  processes,  a  few  simple 
fractions,  antf  United  States  money;  but  just 
there  he  will  be  held  definitely  responsible  for  a 
very  considerable  number  of  the  addition  com 
binations.  The  pupil  may  have  had  fractions  in 
every  grade,  but  the  fifth  grade  will  be  respon- 
sible for  a  thorough  and  systematic  mastery  of 
the  same.  Such  is  the  mixed  method  of  arrange- 
ment which  is  to-day  prevalent  in  American 
schools. 


THE  DISTRIBUTION  OF  OBJECTIVE  WORK 

Objective  Teaching  is  Generally  Current 

THE  use  of  objects  in  teaching  arithmetic  is  cur- 
rent in  the  elementary  school.  Particularly  is 
this  true  in  the  lowest  grades  of  the  school,  in 
primary  work.  It  may  be  said  that  there  is  a  very 
large  quantity  of  objective  teaching  in  the  first 
year  of  schooling  and  that  it  decreases  more  or 
less  gradually  as  the  higher  grades  are  ap- 
proached. By  the  time  the  highest  grammar 
grades  are  reached,  the  use  of  objects  has  reached 
its  minimum. 

The  teaching  of  arithmetic  prior  to  the  middle 
of  the  nineteenth  century  was  little  associated 
with  object  teaching.  That  is  to  say,  the  general 
practice  of  instruction  was  non-objective.  The 
use  of  objects  in  giving  a  concrete  basis  for  ab- 
stract arithmetical  concepts  and  for  memoriter 
manipulations,  seems  to  have  gained  its  initial 
42 


DISTRIBUTION  OF  OBJECTIVE  WORK 

hold  on  the  schools  through  the  introduction  of 
Pestalozzian  methods  of  teaching.  The  later 
introduction  of  school  subjects  requiring  objec- 
tive treatment,  such  as  elementary  science,  na- 
ture study  and  manual  training,  fortified  the 
previous  movement  and  gave  it  considerable 
enlargement.  Together  these  two  movements 
established  the  respectability  of  objective  teach- 
ing in  arithmetic.  School-room  experience  quickly 
gave  it  an  empirical  sanction.  It  remained  for 
the  modern  psychological  movement  in  educa- 
tion to  give  it  a  scientific  sanction,  and  to  refine 
its  uses. 

Its  Distribution  is  Crudefe*  Gauged 

It  is  quite  fair  to  say  that  the  use  of  objective 
work  decreases  more  or  less  gradually  from  the 
first  to  the  last  year,  the  underlying  assumption 
being  that  the  use  of  objects  has  a  teaching 
value  that  decreases  as  the  maturity  of  the  pupils 
increases.  Current  practice  does  not  proceed  far 
beyond  the  application  of  the  simple  and  some- 
what crude  psychological  statement  that  the 
youngest  children  must  have  much  objective 
43 


TEACHING  PRIMARY  ARITHMETIC 

teaching,  the  older  less,  the  oldest  least  of  all. 
The  lack  of  a  more  refined  analysis  of  the  worth 
of  object  teaching  necessarily  leads  to  some 
neglect  and  waste. 

If  a  new  topic  enters  late  into  the  course  of 
study,  as  in  the  case  of  square  root,  the  subject 
is  not  so  well  taught  because  of  the  current  pre- 
judice or  tradition  against  the  use  of  object 
teaching  in  the  higher  grades.  On  the  other 
hand  it  is  also  probable  that  the  teaching  of  ad- 
dition is  often  accompanied  by  wasted  time  and 
energy  simply  because  lingering  over  objects 
in  the  lower  classes  is  the  current  fashion.  • 

Tendency  toward  a  More  Refined  Correlation  of 
Object  Teaching  with  Particular  Immaturity 

Reform  in  the  direction  of  a  more  refined 
and  exact  use  of  object  teaching  has  already 
appeared  in  the  treatment  of  fractions  and  men- 
suration, where,  regardless  of  the  increased 
maturity  of  the  children  studying  these  topics, 
a  large  amount  of  objective  method  is  utilized. 
This  is  a  considerable  departure  from  the  slight 
objective  treatment  of  other  arithmetic  topics 
44 


DISTRIBUTION  OF  OBJECTIVE  WORK 

taught  in  the  same  grades.  Such  exceptional 
practices  suggest  that  the  novelty  of  an  arith- 
metic topic  is  the  condition  calling  for  objec- 
tive work  in  instruction.  I  It  is  immaturity  in 
a  special  subject  or  situation  which  determines 
the  amount  of  basal  objective  work.\  The  jx»r- 
relation  is  not  with  the  age  of  the  pupil,  but 
with  his  experience  with  the  social  problem  or 
subject  in  hand.  It  is  of  course  true  that  the 
younger  the  student  is,  the  greater  the  likelihood 
that  any  subject  presented  will  be  novel  and 
strange.  Only  in  this  indirect  manner  does  the 
novelty  of  subject  matter  coincide  with  mere 
youth  as  an  essential  principle  in  determining 
the  need  of  objective  presentation.  The  naive 
assumption  of  the  older  enthusiastic  reformers 
that  objective  work  is  a  good  thing  psycho- 
logically, one  of  which  the  pupil  cannot  have  too 
much,  is  by  no  means  the  accepted  view  of  the/ 
new  reformer.  With  the  latter,  objective  pre^j 

"V  ~~ ~ "^ 

sentation  is  an  excellent  method  at  a  given  stage 
of  immaturity  in  the  special  topic  involved ;  but 
it  may  be  uneconomical,  even  an  oBstacle^to) 
efficiency,  if  pushed  beyond. 
45 


TEACHING  PRIMARY  ARITHMETIC 

The  Movement  Supported  by  both  Scientific  and 
Common- Sense  Criticism 

There  is,  then,  a  certain  coincidence  of  the  sci- 
entific criticism  of  the  psychologist  and  of  the 
common-sense  criticism  of  the  conservative 
teacher,  who  look  suspiciously  upon  a  highly  ex- 
tended object  teaching.  The  teacher,  on  grounds 
of  experience,  says  that  too  much  objective 
teaching  is  confusing  and  delays  teaching.  The 
psychological  critic  says  it  is  unnecessary  and 
wasteful.  The  result  is  that,  in  these  later  days, 
the  distribution  of  objective  work  has  changed 
somewhat.  More  subjects-are  developed  in  the 
higher  grades  through  an  objective  instruction 
than  before.  Perhaps  no  fewer  subjects  in  the 
lower  grades  are  presented  objectively,  but  the 
extent  of  objective  treatment  of  each  of  these 
has  undergone  considerable  curtailment. 


VI 

THE  MATERIALS  OF  OBJECTIVE  TEACHING 

The  Indiscriminate  Use  of  Objects 

THE  existing  defects  in  objective  teaching  are 
not  restricted  to  a  false  placing  or  distribution. 
The  quality  of  the  teaching  use  of  objects  is 
likewise  open  to  serious  criticism.  Object  teach- 

aoykA*. 

ing  is  a  device,  so  successful,  as  against  prior 
non-objective  teaching,  that  it  has  come  to  be  a 
standard  of  instruction  as  well  as  a  means.  As 
long  as  objects  —  any  convenient  objects  —  are 
used,  the  teaching  is  regarded  as  good.  Given 
such  a  sanction,  the  inevitable  result  is  an  un- 
discriminating  use  of  objects.  The  process  of 
objectifying  tends  not  to  be  regulated  by  the 
needs  of  the  child's  thinking  life ;  it  is  determined 
by  the  -enthusiasm  of  the  teacher  and  the  ma- 
terials convenient  for  school  use. 

The  Artificiality  of  Materials  Utilized 
The  first  fact  which  is  noted  in  observing  ob- 
jective teaching  is  the  artificiality  of  the  materials 
47 


TEACHING  PRIMARY  ARITHMETIC 

employed.  Primary  children  count,  add,  etc.,  with 
things  they  will  never  be  concerned  with  in  life. 
Lentils,  sticks,  tablets,  and  the  like  are  the  stock 
objective  stuff  of  the  schools,  and  to  a  consider- 
able degree  this  will  always  be  the  case.  Cheap 
and  convenient  material  suitable  for  individual 
manipulation  on  the  top  of  a  school  desk  is  not 
plentiful.  But  instances  where  better  and  more 
normal  material  has  been  used  are  frequent 
enough  in  the  best  schools  to  warrant  the  belief 
that  more  could  be  done  in  this  direction  in  the 
average  classroom.  The  "  playing  at  store,"  the 
use  of  actual  applications  of  the  tables  of  weights 
and  measures  are  cases  that  might  be  cited. 

Narrowness  in  the  Range  of  Materials 

The  materials  used  are  not  only  more  artificial 
than  they  need  be,  but  too  restricted  in  range. 
As  has  already  been  said,  the  types  of  material 
capable  of  convenient  and  efficient  use  in  a 
schoolroom  are  not  numerous.  But  the  series 
can  and  ought  to  be  extended.  More  forms  of 
even  the  artificial  material  should  be  used,  thus 
minimizing  the  danger  of  monotony.  The  blame 
48 


MATERIALS  OF  OBJECTIVE  TEACHING 

for  the  narrow  range  of  materials  used  falls  partly 
on  school  boards  who  do  not  vote  a  sufficient 
allowance  for  teaching  materials  to  primary 
teachers  ;  partly  on  teachers  who  do  not  exercise 
sufficient  ingenuity  in  devising  new  forms  of 
objects,  or  who  do  not  show  the  vigor  requisite 
to  a  shift  from  one  material  to  another;  and 
partly  on  the  supervisory  staff  which  has  neither 
been  insistent  upon,  nor  sensitive  to,  the  need 
of  a  more  interesting  range  of  objective  stuffs. 

Inadequate  Variation  of  Traditional  Materials 

Even  the  narrow  range  of  materials  in  general 
use  might  be  better  employed  than  it  is.  There 
is,  of  course,  a  distinct  tendency  to  vary  the  ob- 
jects, merely  because  a  child  gets  tired  of  one 
kind  as  a  material.  But  a  different  quality  of 
variation  is  required  when  the  pupil  is  to  derive 
abstract  notions  from  concrete  materials.  It  is 
too  frequently  the  case  that  the  teacher  will 
treat  the  fundamental  addition  combinations  with 
one  set  of  objects,  e.g.,  lentils.  In  all  the  child's 
objective  experience  within  that  field  there  are 
two  persistent  associations  —  "  lentils  "  and  "  the 
49 


TEACHING  PRIMARY  ARITHMETIC 

relation  of  addition."  The  accidental  element  is 
thus  emphasized  as  frequently  as  the  essential 
one  and,  being  concrete,  has  even  a  better  chance 
to  impress  itself.  A  wide  variation  in  the  objective 
material  used  would  make  teaching  more  effective, 
particularly  with  young  children. 

The  Restricted  Use  of  Diagrams  and  Pictures 

The  nature  of  the  materials  proper  to  objective 
teaching  has  likewise  been  too  narrowly  inter- 
preted. Objective  teaching  has  meant,  almost 
exclusively,  instructing  or  developing  through 
three-dimensional  presentations.  There  is  a  wide 
range  of  two-dimensional  representations  which 
have  been  neglected,  but  which  for  all  the  psy- 
chological purposes  of  education  have  as  much 
worth  as  so-called  objects.  I  refer  here  to  the 
use  of  such  material  as  pictures.  Such  quasi-ob- 
jective material  has  been  little  used  by  teachers 
save  as  it  appears  in  textbooks.  Even  the  text- 
book writers  have  not  used  pictures  with  a  deep 
sense  of  their  intrinsic  worth.  They  are  printed 
as  a  mere  substitute  for  objects  in  a  period  when 
objects  are  popular  pedagogical  materials.  The 
50 


MATERIALS  OF  OBJECTIVE  TEACHING 

geometric  figure  or  diagram  has  had  a  slight  use 
with  both  the  teacher  and  the  textbook  writer. 
Its  most  frequent  use  has  been  in  treatments  of 
mensuration.  There  are,  of  course,  obvious  dis- 
advantages to  pictures  and  diagrams.  The  things 
represented  in  them  are  not  capable  of  personal 
manipulation  by  the  child  in  the  ordinary  sense. 
But.  they  have  a  superiority  all  their  own.  They 
offer  a  wider,  more  natural,  and  more  interesting 
range  of  concrete  experiences. 

Plays  and  Games  in  Object  Teaching 

There  are  other  curious  phases  of  narrowness 
in  the  current  pedagogical  interpretation  of  what 
constitutes  a  concrete  or  objective  experience. 
It  will  be  noted  that  visual  objects  are  the  ones 
generally  employed  and  that  they  are  generally 
inanimate  objects.  Of  late  there  has  been  some 
tendency  to  use  hearing  and  touch  in  giving  a 
concrete  basis  to  teaching.  Advantage  is  taken 
of  the  social  plays  of  children,  and  their  games 
with  things.  Here  the  children  themselves,  and 

their  relations  and  acts  are  the  experiences  from 

• 

which  the  numerical  units  are  obtained.   With 
Si 


TEACHING  PRIMARY  ARITHMETIC 

some  of  the  best  teachers  in  the  lowest  grades 
it  is  no  longer  unusual  to  see  children  moving 
about  in  all  sorts  of  play  designed  to  add  reality 
to,  and  increase  interest  in,  number  facts. 

The  Lack  of  Unity  in  the  Use  of  Objects 

The  conservative  teacher's  use  of  objects  is 
hopelessly  artificial  and  lacking  in  unity.  If  he 
brings  a  series  of  objects  into  the  development 
of  a  single  topic,  they  have  little  relation  to  each 
other,  and  they  represent  no  actual  grouping. 
Their  sole  connection  with  one  another  is  that 
they  exemplify  the  same  abstract  arithmetical 
truth.  Beans,  cardboard  squares,  and  shoe-pegs 
may  all  be  employed  in  the  same  lesson.  The 
progressive  teachers  offer  more  logical  unity  in 
their  materials.  To  "play  at  store,"  to  utilize 
games,  to  deal  with  things  within  a  picture,  is  to 
bring  the  concrete  materials  into  the  classroom 
with  a  more  nearly  normal  setting.  It  is  in  no 
small  measure  due  to  this  better  use  of  material 
that  the  progressive  teacher  is  gaining  power 
throughout  the  elementary  grades. 


VII  y  3 


SOME  RECENT  INFLUENCES  'ON  OBJECTIVE 
TEACHING 

THE  wasteful  use  of  objective  teaching  in  the 
lowest  grades  has  undergone  some  correction. 
The  sheer  enthusiasm  of  the  modern  reformer  is 
partly  responsible  for  this  modification  of  con- 
servative practice.  When  did  single-minded  men 
ever  keep  within  bounds  ?  In  our  social  economy 
the  defense  of  the  radical  is  found  in  the  fact 
that  other  single-minded  men  are  conservatives. 
Men  at  one  extreme  need  to  be  overcome  by  like 
men  at  the  other.  But  the  check  of  one  enthu- 
siast on  another  is  not  always  perfect.  Other 
contributory  causes  make  it  easy  to  go  to  unfor- 
tunate extremes  not  easily  corrected. 

The  Influence  of  Inductive  Teaching 

Inductive  teaching  has  been  one  of  several 
movements  affecting   objective   teaching.   The 
effort  of  teachers  to  escape  the  slavishness  of 
53 


TEACHING  PRIMARY  ARITHMETIC 

mere  memoriter  methods  and  to  approximate 
real  thinking  led  to  the  introduction  of  inductive 
teaching.  Necessarily  objective  teaching  became 
more  or  less  identified  with  the  new  movement 
and  was  influenced  by  it  So,  it  has  been  said 
of  objective  work  in  arithmetic  as  it  has  been 
said  of  laboratory  work  in  the  sciences,  that 
such  instruction  is  a  method  of  "  discovery  "  or 
"rediscovery."  Such  an  alliance  has  had  its  ben- 
eficial effects  upon  objective  teaching ;  it  has  re- 
deemed it  from  the  aimless  "observational  work  " 
of  an  earlier  "  objective  study."  But  in  the  teach- 
ing of  arithmetic,  at  any  rate,  it  has  confused  an 
objective  mode  of  presentation  with  a  scientific 
method  of  learning  truth,  two  activities  having 
a  common  logical  basis,  but  not  at  all  the  same. 
Under  the  assumption  that  the  "  development " 
method  is  one  of  "rediscovery,"  the  tendency  is 
to  give  the  child  as  complete  a  range  of  con- 
crete evidences  as  would  be  necessary  on  the 
part  of  the  scientist  in  substantiating  a  new  fact. 
The  result  is,  that  long  after  the  child  is  con- 
vinced of  the  truth,  say  that  4  and  2  are  6,  the 
teacher  persists  in  giving  further  objective  illus- 
54 


INFLUENCES  ON  OBJECTIVE  TEACHING 

trations  of  the  fact.  The  child  loses  interest  in 
the  somewhat  monotonous  continuance  of  ob- 
jective manipulations,  and  the  teacher  has  natu- 
rally wasted  time  and  energy.  If  the  fact  or  the 
process  that  the  teacher  wishes  to  convey  can 
be  transmitted  with  fewer  objective  treatments 
(the  authoritative  treatment  of  the  teacher  count- 
ing for  something  in  school,  as  authority  counts 
everywhere),  then  it  is  unnecessary  to  exhaust 
the  objective  treatments  of  a  numerical  fact.  In- 
ductive teaching  and  learning  are  not  equivalent 
to  inductive  discovery ;  and  to  hold  them  iden- 
tical is  necessarily  to  overdo  the  use  of  objects  in 
teaching. 

The  Movement  for  Active  Modes  of  Learning 

Another  modern  movement  in  teaching  method 
which  has  had  a  conspicuous  effect  on  objective 
teaching  is  the  Froebelian  demand  for  "self -ac- 
tivity ".on  the  part  of  the  child.  The  recent  favor 
enjoyed  by  manual  training,  nature  study,  self- 
government,  and  other  active  phases  of  school 
life  is  indicative  of  the  sway  of  this  doctrine.  Its 
influence  has  forced  the  introduction  of  new  sub- 
55 


TEACHING  PRIMARY  ARITHMETIC 

jects  and  changed  the  manner  of  presenting  the 
older  subjects  of  the  elementary  curriculum. 
Arithmetic  has  responded  along  with  the  other 
studies,  and  an  active  use  of  objects  by  the  chil- 
dren themselves  is,  found  in  increased  degree. 

There  was  a  time  when  objective  work  in  the 
schools  was  a  passive  matter  so  far  as  the  child 
was  concerned.  Any  active  manipulation  of  the 
objects  that  might  be  required  was  cared  for  by 
the  teacher,  the  child  being  merely  a  passive 
observer.  This  is  much  less  the  case  than  for- 
merly, the  influence  of  "self-activity"  having 
entered  with  contemporaneous  pedagogy.  The 
present  situation  is  one  where  the  child  some- 
times merely  observes  objects  and  sometimes 
actually  handles  them. 

At  present,  then,  we  have  about  the  same 
range  of  methods  employed  in  teaching  arith- 
metic as  in  teaching  science.  At  one  extreme 
the  teacher  himself  demonstrates  by  the  help  of 
objects  in  the  presence  of  the  class,  and  records 
the  relations  in  appropriate  arithmetical  symbols, 
the  class  being  in  the  position  of  interested  spec- 
tators of  a  process.  At  the  other  extreme  the 
56 


INFLUENCES  ON  OBJECTIVE  TEACHING 

teacher  puts  the  material  on  the  desks  of  the 
children  and,  with  a  minimum  of  instruction  in 
advance,  directs  them  toward  the  desired  expe- 
riences and  conclusions. 

The  Abbreviated  Use  of  Objects 

As  might  be  expected,  there  has  been  some 
reaction  against  the  influences  emerging  from 
inductive  or  developmental  teaching  and  active 
modes  of  learning.  To  a  considerable  degree  the 
reactionary  influence  expresses  itself  in  the  ab- 
breviated use  of  objects  in  presenting  a  mathe- 
matical relation,  process,  or  manipulation.  One 
mode  of  abbreviation  will  suffice  as  an  example. 

There  are  two  methods  of  relating  the  sym- 
bols and  processes  of  arithmetic  to  the  actual 
relations  among  objects.  For  convenience  these 
maybe  called  the  methods  of  "parallel  corre- 
spondence" and  of  "final  correspondence." 

The  Method  of  Parallel  Correspondence 

The  method  of  "  parallel  correspondence  "  is 
generally  used  in  the   development  of  all  the 
simpler  combinations  or  processes  of  arithmetic. 
57 


TEACHING  PRIMARY  ARITHMETIC 

In  learning  to  count,  the  child  sees  the  first  ob- 
ject and  says  the  symbolic  "  one,"  sees  the  sec- 
ond object,  and  says  the  symbolic  "two."  Again 
in  addition,  he  sees  "  ten,"  and  writes  the  sym- 
bol 10 ;  sees  "  six,"  and  writes  6 ;  sees  the  whole 
as  sixteen,  and  writes  16  ;  then  summarizes  the 
work  in  the  form  10  +  6=  16.  Each  stage  in  the 
symbolic  process  is  noted  in  connection  with  ob- 
jects. This,  the  method  of  "parallel  correspond- 
ence," is  the  more  current  method  of  using 
objects. 

The  Method  of  Final  Correspondence 

The  method  of  showing  a  "final  correspond- 
ence" of  result  between  objective  manipulation 
and  symbolic  manipulation  is  much  less  fre- 
quently used.  It  is  used  with  more  complex  pro- 
cesses than  those  mentioned  above,  in  connec- 
tion with  column  addition  or  "borrowing"  in 
subtraction.  It  is  a  mode  of  object  teaching  used 
in  place  of  the  usual  "explanation"  or  "ration- 
alization "  which  attempts  to  explain  what  is  sim- 
ply a  correspondence  between  the  manipulation 
of  a  series  of  facts  and  the  manipulation  of  a  se- 
58 


INFLUENCES  ON  OBJECTIVE  TEACHING 

ries  of  symbols.  Under  this  method  the  teacher 
usually  tells  the  child  directly  how  to  perform 
the  process  in  the  conventional  manner,  no  spe- 
cial explanation  being  given.  Then  a  case  in- 
volving the  actual  use  of  objects  is  considered, 
and  this  result  is  compared  with  the  result  ob- 
tained by  the  symbolicv  manipulation.  One  or 
two  such  cases  suffice  to  convince  the  pupil  that 
the  authoritative  mode  is  true  to  nature.  This 
method  of  "  final  correspondence  "  in  the  use  of 
objects  represents  a  new  and  restricted,  but  in- 
creasing, tendency. 


VIII 

THE  USE  OF  METHODS  OF  RATIONALIZATION 

The  Tendency  toward  Rational  Methods 
SOME  of  the  marked  changes  which  have  occurred 
in  the  methods  employed  for  the  presentation  of 
number  to  children  have  already  been  mentioned 
in  connection  with  the  objective  teaching  of 
arithmetic.  The  main  tendency  to  be  noted  is 
that  objective  instruction,  which  has  been  used 
as  a  mere  device  of  illustration,  becomes  the 
first  step  in  inductive  or  developmental  teaching. 
It  is  subsumed  under  a  more  inclusive  method. 
The  change  is  significant,  for  it  is  a  symp- 
tom indicating  that  mathematical  teaching  is 
becoming  less  dominantly  memoriter  and  more 
rational. 

The  Era  of  Direct  Instruction  and  Drill 

Several  decades  ago  it  was  not  at  all  unusual 
for  the  bare  facts  of  arithmetic  to  be  given  to 
the  child  by  the  teacher  without  much  attempt  at 
60 


METHODS  OF  RATIONALIZATION 

providing  a  basis  in  the  pupil's  own  experience. 
The  teaching  was  "direct,"  the  teacher's  atten4 
tion  being  focused  on  getting  the  fact  from  his) 
own  mind  into  the  child's  mind,  -the  whole  env 
phasis  being  placed  upon  the  problem  of  trans- 
mission and  the  subsequent  difficulty  of  retention. 
In  so  far  as  objects  and  illustrations  were  used, 
they  were  merely  incidental  and  reinforcive.  They 
did  not  constitute  any  basic  body  of  personal  ex- 
perience by  which  the  child  was  to  seize  a  con- 
cept, relation,  or  process  to  be  handled  thereafter 
through  symbols  and  conventional  forms.  Under 
such  a  system  of  instruction,  still  too  widely  prev- 
alent, the  child  had  to  memorize  outright,  with- 
out any  concrete  basis  for  his  belief,  tables  of 
addition,  multiplication,  etc.,  and  the  rules  for 

solving  various  types  of  problems.  It-was  outright 

* 

memorization  for  which   little  vital  motivation 
was  provided.  In  insuring  retention  the  teacher 

therefore  relied,  not  upon  interested  a'nd  varied 

•'.*.' 
impressions,  but  upon  the  number  of  verbal  re- 

petitions. "  Drill  "  was  characteristic  of  this  era 
in  teaching. 


61 


\, 


TEACHING  PRIMARY  ARITHMETIC 

Indirect  Teaching  as  a  Rational  Method 

Under  the  influence  of  the  inductive  or  de- 
velopmental method  of  teaching,  the  emphasis 
on  the  repetitional  memorization  of  number  facts 
and  processes  is  reduced.  Teaching  now  becomes 
"indirect"  rather  than  "direct"  ;  the  child  learns 
through  his  own  experience  rather  than  through 
the  statement  of  book  or  teacher.  Here  the  child's 
own  thought  and  activity,  not  the  teacher's,  are 
conspicuously  central  in  the  teaching  situation. 
The  teacher  stimulates  the  child  into  action  ;  he 
suggests,  guides,  corrects,  does  everything  in  fact 
save  obtrude  his  authority  and  opinion  into  the 
child's  interpretation.  The  child's  activity  gives 
him  many  vivid  and  varied  impressions  of  the  sub- 
traction combination  or  other  relation.  When  he 
has  found  the  fact,  he  has  already  learned  it ! 
Further  drill  or  review  is  not  primary,  but  simply 
supplementary — a  further  guarantee  of  the  per- 
sistence of  the  impression.  Even  the  spirit  of 
such  supplementary  drill  and  review  is,  in  these 
days,  something  different  from  a  monotonous  re- 
petition of  the  same  words  ;  it  is  a  reimpression 
62 


METHODS  OF  RATIONALIZATION 

or  review  of  the  essential  fact  in  many  varied  and 
interesting  forms. 

Interest  as  a  Factor  in  Methods  of  Rational- 
ization 

It  is  perfectly  natural  that,  in  shifting  the 
teacher's  attention  from  his  own  activities  to  those 
of  the  children,  the  interest  of  the  child  should 
be  considered  in  increasing  degree.  If  the  child 
is  to  learn  directly,  with  a  maximum  use  of  his 
initiative,  it  is  absolutely  essential  that  the 
teacher  should  provide  some  motive.  This  im- 
plies that  the  child  is  to  be  interested  in  some 
fundamental  way  in  the  activities  in  which  he  is 
to  engage.  Instead  of  thumbing  the  fundamental 
facts  with  his  memory,  in  an  artificial  and  effort- 
ful manner,  "sing-songing"  the  tables  rhythmi- 
cally, so  as  to  make  dull  business  less  dull,  the 
child  studies  the  arithmetic  involved  in  his  own 
life,  for  the  modern  teacher  brings  the  two  to- 
gether. The  number  story,  the  arithmetical  game, 
playing  at  adult  activities,  constructive  work, 
measuring,  and  other  vital  interests  of  the  child 
and  community  life  become  increasingly  the  basis 
63 


TEACHING  PRIMARY  ARITHMETIC 

of  instruction  in  number.  Such  is  the  pronounced 
tendency  wherever  the  movement  is  away  from 
the  traditional  rote-learning  or  drill. 

Of  course  there  is  a  slight  tendency  in  Amer- 
ican elementary  schools,  where  a  soft  and  false 
interpretation  of  the  "doctrine  of  interest"  is 
gospel,  to  teach  only  those  things  that  can  be 
taught  in  an  interesting  fashion.  But  this  tend- 
ency is  less  operative  in  arithmetic  than  in 
other  subjects.  Here  the  logical  interdependence 
of  one  arithmetical  skill  on  another  has  quickly 
pointed  the  failure  of  such  a  haphazard  mode  of 
instruction. 

The  Reaction  against  Rationalization 

There  is,  however,  in  "  advanced  "  as  well  as 
in  reactionary  quarters,  a  revolt  against  the  tend- 
ency to  objectify,  explain,  or  rationalize  every- 
thing taught  in  arithmetic.  On  the  whole  it  is  a 
discriminating  movement,  for  this  opposition  to 
"rationalization"  in  arithmetical  teaching,  and 
in  favor  of  "memorization"  or  "  habituation," 
bases  its  plea  on  rational  grounds,  mainly  derived 
from  the  facts  of  modern  psychology. 
64 


METHODS  OF  RATIONALIZATION 

It  is  specifically  opposed  to  explaining  why 
"carrying"  in  addition,  and  "borrowing"  in  sub- 
traction are  right  modes  of  procedure.  These 
acts  are  to  be  taught  as  memory  or  habit,  inas- 
much as  they  are  to  be  performed  by  that  method 
forever  after.  To  develop  such  processes  ration- 
ally or  to  demand  a  reason  for  the  procedure  once 
it  is  acquired,  is  merely  to  stir  up  unnecessary 
trouble,  trouble  unprompted  by  any  demands  of 
actual  efficiency. 

Four  Principles  for  the  Use  of  Rationalization 

A  study  of  the  actual  arithmetical  facts  upon 
which  this  opposition  expresses  itself  suggests 
the  four  following  general  principles  as  to  the 
use  of  "rationalization"  and  " habituation,"  as 
methods  of  mastery:  (i)  Any  fact  or  process 
which  always  recurs  in  an  identical  manner,  and 
occurs  with  sufficient  frequency  to  be  remem- 
bered, ought  not  to  be  "rationalized"  for  the 
pupil,  but  "habituated."  The  correct  placing 
of  partial  products  in  the  multiplication  of  two 
numbers  of  two  or  more  figures  is  a  specific  case. 
(2)  If  a  process  does  recur  in  the  same  manner, 
65 


TEACHING  PRIMARY  ARITHMETIC 

but  is  so  little  used  in  after  life  that  any  formal 
method  of  solution  would  be  forgotten,  then  the 
teacher  should  "rationalize"  it.  The  process  of 
finding  the  square  root  of  a  number  illustrates 
this  series  of  facts.  (3)  If  the  process  always  does 
occur  in  the  same  manner,  but  with  the  frequency 
of  its  recurrence  in  doubt,  the  teacher  should 
both  "habituate"  and  "  rationalize."  The  division 
of  a  fraction  by  a  fraction  is  frequently  taught 
both  "mechanically"  and  "by  thinking  it  out." 
(4)  When  a  process  or  relation  is  likely  to  be  ex- 
pressed in  a  variable  form,  then  the  child  must 
be  taught  to  think  through  the  relations  involved, 
and  should  not  be  permitted  to  treat  it  mechan- 
ically, through  a  mere  act  of  habit  or  memory. 
All  applied  examples  are  to  be  dealt  with  in  this, 
manner,  for  such  problems  are  of  many  types, 
and  no  two  problems  of  the  same  type  are  ever 
quite  alike.  These  laws  will,  of  course,  not  be 
interpreted  to  mean  that  no  reason  is  to  be  given 
a  child  in  a  process  like  "  carrying  "  in  addition. 
The  reason  is  not  essential  to  efficient  mastery, 
but  it  may  be  given  to  add  interest  or  to  satisfy 
the  specially  curious. 

66 


METHODS  OF  RATIONALIZATION 

The  Substantiating  Psychology 
The  theoretic  basis  which  seems  to  underlie 
such  a  statement  of  general  principles  is  derived 
from  functional  psychology.  Memory  and  reason- 
ing are  not  separate  functions  ;  they  are  interde- 
pendent ;  but  we  mean  differently  by  these  terms 
because  they  have  distinguishable  emphases.  It 
may  be  said  that  memory  as  a  function  is  effi- 
cient only  in  the  face  of  familiar  situations 
where,  if  the  association  is  present  at  all,  the 
response  is  quick  and  precise.  In  the  face  of 
new  situations  it  is  incapable  of  accurate  re- 
sponse. Reason  is  slow  and  uneconomical  in 
action,  but  it  is  the  only  efficient  method  of 
arriving  at  the  essential  nature  of  a  problem 
largely  unfamiliar.  It  would  be  wasteful  to  meet 
many  of  the  necessary  events  of  life  with  a 
purely  reasoned  reaction.  It  would  be  too  de- 
vious and  deliberate  in  reaching  conclusions. 

Rationalization  as  a  Substititte  for  Object 

Teaching 

There  is  a  sense  in  which  all  proof  through 
objects  is  a  type  of  rationalization,  but  we  do 

67 


TEACHING  PRIMARY  ARITHMETIC 

not  ordinarily  so  consider  it.  Such  a  mere  "cor- 
respondence with  the  objective  facts  "  is  suffi- 
ciently different  from  the  method  of  "  explaining 
a  new  fact  in  terms  of  previously  acquired  facts" 
to  warrant  a  separate  classification.  Were  it 
not  that  the  methods  are  sometimes  inter- 
changeable in  developing  arithmetical  truths, 
they  would  not  need  to  be  mentioned  here.  A 
citation  will  make  the  point  clear.  In  teaching 
the  multiplication  tables,  the  combination  "  six 
twos  are  twelve  "  may  be  taught  as  a  direct  ob- 
jective fact,  as  when  six  pupils  with  two  hands 
each  are  shown.  On  the  other  hand  the  same 
fact  of  multiplication  may  be  taught  as  a  "  de- 
rived fact,"  as  when  six  twos  are  added  in  a 
column  and  make  twelve.  They  are  both  rational 
methods  of  proving  that  six  twos  are  twelve. 
One  method  shows  it  objectively,  the  other 
through  the  use  of  well  established  addition 
combinations  viewed  as  multiplication.  Such  a 
rational  mode  of  "deriving"  multiplication  is 
used  more  frequently  perhaps  than  the  objective 
method. 


IX  / 

SPECIAL    METHODS    FOR    OBTAINING    ACCU- 
RACY, INDEPENDENCE,  AND   SPEED 

Supervision  of  Learning  after  First  Development 

IT  is  not  alone  the  first  stages  in  the  acquisition 
of  an  arithmetical  process  which  have  received 
attention  in  the  re-organization  of  teaching 
methods,  though,  to  be  sure,  the  problem  of 
first  presentations  has  in  recent  decades  been 
given  the  most  attention.  More  and  more,  the 
American  tendency  is  to  watch  every  step  in 
the  learning  process,  to  provide  for  all  necessary 
transitions,  and  to  safeguard  against  avoidable 
confusions.  It  might  be  suggested  that  the  in- 
termediation of  the  teacher  at  every  step  in  the 
child's  work  might  destroy  the  pupil's  initiative 
and  independence.  Apparently,  however,  those 
who  are  deeply  interested  that  the  child  should 
not  be  permitted  to  fall  into  the  errors  which 
unsupervised  drill  would  convert  into  habits,  are 
fully  as  cautious  to  provide  steps  for  forcing  the 
69 


TEACHING  PRIMARY  ARITHMETIC 

child  to  assume  an  increasing  responsibility  for 
his  own  work.  The  distinction  made  is  that  an 
over-early  independence  is  as  fatal  to  accurate 
and  rapid  mathematical  work  as  an  over-delayed 
dependence. 

The  Use  of  Steps,  or  Stages,  in  Teaching 

Some  of  these  serial  treatments  or  related 
stages  of  method  to  which  reference  has  been 
made  may  be  cited.  Necessarily  only  the  more 
important  are  mentioned.  In  indicating  certain 
clean-cut  steps  or  processes,  there  is  no  intent  to 
convey  the  idea  that  these  stages  are  fixed  or 
conscious  matters.  The  statement  is  merely  in- 
dicative of  the  habitual  tendency  of  the  average 
practitioner  with  an  implied  theory.  As  will  be 
readily  evident,  there  is  no  assumption  that  such 
a  formal,  classified,  theoretic  statement  of  stages 
is  a  conscious  possession  of  the  teaching  staff  in 
general.  Again,  in  actual  school  work  there  are 
many  types  of  variation  from  the  characteristic 
modes  here  suggested.  Always  the  steps  pverlap ; 
frequently  they  are  extended,  abbreviated,  or 
omitted.  But  the  statement  represents,  in  a  very 
70 


SPECIAL  METHODS 

real  way,  the  trend  of  underlying  theory,  whether 
conscious  or  merely  implied. 

Stages  in  the  Presentation  of  Problems 

One  of  the  specific  controversies  much  argued 
in  the  primary  school  concerns  the  medium 
through  which  arithmetical  examples  and  prob- 
lems shall  be  transmitted  to  young  children. 
There  are  three  typical  ways  in  which  a  situa- 
tion demanding  arithmetical  solution  may  be 
brought  to  the  child's  mind :  (i)  The  situation 
when  visible  may  be  presented  through  itself, 
that  is,  objectively.  (2)  The  situation  may  be 
described  through  the  medium  of  spoken  lan- 
guage, the  teacher  usually  giving  the  dictation. 
(3)  The  situation  may  be  conveyed  through 
written  language,  as  when  the  child  reads  from 
blackboard  or  text.  Inasmuch  as  objects  are  a 
universal  language,  no  difficulty  arises  through 
this  basic  method  of  presentation.  It  is  when  a 
language  description  of  a  situation  is  substituted 
for  the  situation  itself  that  difficulty  occurs.  The 
child  might  be  able  to  solve  the  problem  if  he 
really  understood  the  situation  the  language  was 
71 


TEACHING  PRIMARY  ARITHMETIC 

meant  to  convey.  Owing  to  the  difficulty  that 
primary  children  have  in  getting  the  thought  out 
of  language,  it  has  been  urged  that  problems 
in  any  unfamiliar  field  should  be  presented  in 
the  following  order :  (i)  Objectively  or  graphi- 
cally ;  (2)  when  the  fundamental  idea  is  grasped, 
through  spoken  language  ;  and  (3),  after  the  type 
of  situation  is  fairly  familiar,  through  written  or 
printed  language.  It  is  seriously  urged  by  some 
teachers  that  no  written  presentation  should  be 
used  in  the  first  four  grades.  Such  an  extreme 
tendency  would  practically  abolish  the  use  of 
primary  text-books.  There  are  many  exceptional 
teachers  who  do  not  put  a  primary  text  in  the 
hands  of  children  at  all.  Such  a  tendency  is  in- 
creasing. Particularly  is  this  true  among  primary 
teachers  in  the  schools  of  the  foreign  quarters  of 
large  cities.  Accurate  communication  through 
the  English  language  is  always  more  difficult  here. 
Hence,  the  period  of  objective  teaching  is  neces- 
sarily prolonged,  dependence  on  the  "number 
stories  "  told  by  the  teacher  extended,  and  the 
solution  of  written  problems  much  longer  de- 
layed than  elsewhere. 

72 


SPECIAL  METHODS 

An  Opposite  Method  in  Presenting  Examples 

The  situation  is  somewhat  different,  almost 
the  opposite  in  fact,  when  "  examples "  rather 
than  "problems  "  are  presented,  meaning  by  " ex- 
ample" a  "problem  "  expressed  through  the  use 
of  mathematical  signs.  It  is  easier  to  present 
"  examples  "  in  written  form  on  blackboard  or  in 
text  than  it  is  to  dictate  them  orally.  This  ob- 
viates the  necessity  of  holding  the  examples  in 
mind  during  solution.  The  permanence  of  the 
visual  presentation  saves  the  re-statement  fre- 
quently necessary  in  oral  presentation.  Hence  it 
is  a  common  practice  to  supply  the  youngest  chil- 
dren with  mimeographed  or  written  sheets  of  "ex- 
amples." It  is  with  older  children,  or  with  younger 
children  at  a  later  stage  in  the  mastery  of  a  typical 
difficulty,  that  oral  presentation  of  examples  is 
stressed.  Then  we  have  that  type  of  work  which 
is  called  "  mental  "  or  "  silent  "  arithmetic. 

Better   Transitions  from    Concrete  to  Abstract 

Work 

There  is  some  tendency  toward  the  provision 
of  better  transitions  from  the  objective  presenta- 
73 


TEACHING  PRIMARY  ARITHMETIC 

tion  of  applied  problems  to  the  symbolic  pre- 
sentation of  abstract  examples.  Behind  all  uses 
of  objective  work  is  the  belief  that  it  is  a  mere 
foundation  for  more  rapid  and  efficient  abstract 
work.  Objective  teaching  is  fundamental,  but 
purely  preparatory.  The  child  ought  to  pass 
from  objects  and  sense-impressions,  through 
images  of  various  degrees  of  abbreviation,  to 
symbols  and  the  abstract  concepts  for  which 
they  stand.  But  in  American  practice  a  sharp 
jump  is  usually  made  from  concrete  objects  to 
abstract  symbols.  The  transition  through  ade- 
quate transitional  imagery  is  not  made.  Wher- 
ever psychological  influences  are  directly  at  work 
in  the  schools,  there  is  a  minority  movement 
favoring  a  better  transition.  The  nature  of  such 
transition  is  scarcely  reasoned  out  as  so  much 
psychological  science,  but  is  the  accompaniment 
of  a  widening  professional  movement  for  the 
enlarged  use  of  pictures,  diagrams,  number 
stories,  and  the  like.  A  critical  examination  of 
the  various  means  of  presenting  arithmetical 
situations  would  order  them  as  follows  in  making 
the  transitions  from  objective  concreteness  to 
74 


SPECIAL  METHODS 

symbolic  abstraction:  (i)  Objects,  (2)  pictures, 
(3)  graphs,  (4)  the  concrete  imagery  of  words, 
(5)  more  abstract  verbal  presentations,  (6)  pre- 
sentations through  mathematical  symbols.  No 
such  minuteness  of  adjustment  is  apparent  in 
existing  methods,  though  it  might  seem  desir- 
able in  teaching  young  children.  At  any  rate,  it 
would  be  more  effective  than  an  unreasoned  tra- 
ditional procedure  full  of  over-emphasis  and 
omission. 

The  Child's  Four  Modes  of  Work 

We  have  thus  far  discussed  merely  the  teach- 
er's activity  in  instruction.  We  have  to  note  the 
graded  requirement  made  in  the  child's  own  ac- 
tivity. What  is  the  existing  custom  with  refer- 
ence to  the  manner  !n  which  children  are  required 
to  solve  the  problems  or  examples  presented  to 
them  ?  There  are  four  typical  ways  in  which  the 
child  does  his  work,  the  names  of  which  are  de- 
rived from  the  differentiating  element :  (i)  The 
"  silent "  method,  otherwise  spoken  of  as  "  mental 
arithmetic,"  "arithmetic  without  a  pencil,"  etc. 
(2)  The  "oral"  method  where  the  child  works 
75 


TEACHING  PRIMARY  ARITHMETIC 

aloud,  that  is,  expresses  his  procedure  step  by 
step  in  speech.  (3)  The  "  written  "  method  where 
the  child  writes  out  in  full  his  analysis  and  cal- 
culation. (4)  The  "mixed"  method  where  the 
child  uses  all  three  of  the  previously  mentioned 
methods,  in  alternation,  as  ease  and  efficiency 
may  require. 

The  Worth  of  these  Modes 

The  worth  of  these  four  methods  of  work  is 
necessarily  variable.  The  rapidity  of  the  "  silent 
method"  with  simple  figures  is  obvious.  The 
"silent  method  "and  the  "mixed  method"  (which 
is  more  slow  but  more  manageable  with  complex 
processes  and  calculations)  are  the  two  methods 
normally  employed  in  social  and  business  life. 
The  purely  " oral "  and  "written  "  methods,  with 
their  tendency  toward  analysis  and  calculation 
fully  expressed  in  oral  or  written  language,  are 
highly  artificial.  They  are  valuable  as  school 
devices  for  revealing  the  action  of  the  child's 
mind  to  the  teacher  so  that  it  may  be  corrected, 
guided,  and  generally  controlled.  The  present 
tendency  is  toward  an  over-use  of  these  methods 
76 


SPECIAL  METHODS 

and  toward  an  under-use  of  the  other  two,  more 
particularly  the  "  mixed  "  method.  It  would  seem 
that  there  is  little  conscious  attempt  to  make 
certain  that  the  child  moves  from  full  oral  or 
written  statement  to  the  judicious  application  of 
the  more  natural "  silent"  and  "  mixed  "  methods. 
It  may  be  that  full  oral  and  written  statements 
of  work  have  seriously  hampered  the  right  use  of 
the  more  natural  methods  of  statement. 

The  Traditiottal  Quarrel  between  "  Mental "  and 
"  Written  "  Arithmetic 

One  conspicuous  traditional  quarrel  in  the 
schools  is  between  the  "  silent "  and  the  "  writ- 
ten "  methods.  Up  to  within  a  decade  or  so  ago, 
"silent"  or  "mental "  arithmetic  was  much  over- 
emphasized, being  carried  to  absurd  extremes. 
The  reaction  that  followed  was  equally  extreme 
in  its  emphasis  on  the  "  written  "  method.  There 
are  signs  now  of  a  more  rational  use  of  the  two 
as  supplements  of  each  other. 

The  order  in  which  different  statements  of 
arithmetical  work  should  come  has  also  been  a 
subject  for  pedagogical  argument.  The  usual 
77 


TEACHING  PRIMARY  ARITHMETIC 

order,  due  to  the  fact  that  first  treatments  of  a 
topic  are  simple  both  in  the  steps  of  reasoning 
and  the  calculations  involved,  has  been  "  silent " 
arithmetic  followed  by  "  written  "  arithmetic.  A 
more  recent  order  has  been:  (i)  "silent"  (2) 
"  written  "  (3)  "  silent "  —  a  much  superior  serial 
order,  though  by  no  means  an  accurate  state- 
ment of  a  perfect  procedure.  The  fixed  treat- 
ments necessitated  by  text-books  have  made 
teaching  method  arbitrary  in  its  steps,  here  as 
elsewhere.  It  is  altogether  probable  that  many 
simple  calculations  or  analyses  can  be  done 
"silently"  ("mentally")  from  the  beginning; 
that  others  require  visual  demonstration,  but  once 
mastered  can  thereafter  be  done  without  visual 
aids ;  that  still  others  will  always  have  to  be  per- 
formed, partially  at  least,  with  some  written  work. 
It  is  purely  a  matter  for  concrete  judgment 
in  each  special  case,  but  the  existing  practice 
scarcely  recognizes  this  truth.  The  result  is  that 
many  problems  are  arbitrarily  done  in  one  way, 
and  it  is  too  frequently  the  uneconomical  and  in- 
efficient way  that  is  used. 


SPECIAL  METHODS 

The  Transition  from  Development  by  Teacher  to 
Independent  Work  by  Pupil 

It  is  well  to  recall  that  in  all  these  efforts  to 
control  the  child's  activity,  there  is  a  tendency 
to  leave  the  child  over-dependent  upon  the 
teacher.  It  is  vitally  important  that  a  child 
should  be  kept  free  of  any  error  which  unsuper- 
vised  drill  would  fix  into  a  stubborn  habit ;  but 
it  is  likewise  important  that  the  child  should 
acquire  some  self-reliance.  While  not  always 
clearly  defined,  there  is  a  distinct  tendency  in 
the  direction  of  releasing  the  teacher's  control 
of  the  child.  A  characteristic  practice  would  be 
one  in  which  the  teacher's  work  with  the  child 
would  pass  through  various  stages,  each  one  of 
which  would  mark  a  decrease  in  the  control  of 
the  process  by  the  teacher  and  an  increase  in 
the  freedom  of  the  child  to  do  his  example,  or 
problem,  by  himself. 

Four  Characteristic  Stages  of  the  Transition 

One  characteristic  series  of  stages  quite  fre- 
quently used  in  the  presentation  of  a  single  topic 
79 


TEACHING  PRIMARY  ARITHMETIC 

in  arithmetic,  say  "carrying"  in  addition,  is  the 
following:  (i)  The  teacher  performs  the  pro- 
cess on  the  blackboard  in  the  presence  of  the 
class,  the  children  not  being  allowed  to  attempt 
the  process  by  themselves  until  after  the  process 
is  clearly  understood  from  the  teacher's  develop- 
ment. (2)  The  children  are  then  allowed  to  per- 
form the  process  upon  the  blackboard,  where  it 
is  exceedingly  easy  for  the  teacher  to  keep  the 
work  of  every  child  under  his  eye.  An  error  is 
caught  by  a  quick  glance  at  the  board  and  im- 
mediately corrected  before  the  child  can  reiterate 
a  false  impression.  (3)  Other  cases  of  the  same 
type  of  example  are  assigned  to  the  children  at 
their  seats  where  they  work  upon  paper,  still 
under  the  supervision  of  the  teacher  —  a  super- 
vision which  is  less  adequate,  however.  (4)  The 
same  difficulty,  after  the  careful  safeguarding  of 
the  previous  stages,  is  then  assigned  for  "home 
work,"  where  the  child  relies  almost  completely 
upon  himself.  Once  more  it  is  necessary  to  sug- 
gest that  these  stages  are  merely  roughly  im- 
plied in  the  variations  of  existing  practice. 


80 


SPECIAL  METHODS 

Special  Methods  of  Attaining  Speed 

Most  of  the  methods  discussed  in  this  chapter 
have  had  as  their  sanction  the  attainment  of  ac- 
curacy in  thinking  and  calculating.  Some  efforts 
to  insure  independent  power  on  the  part  of  the 
child  have  already  been  noted.  But  nothing  has 
been  said  of  the  effort  to  add  speed  to  accuracy 
in  getting  efficient  results.  Such  special  efforts 
have  been  made.  These  efforts  may  be  classified 
into  two  groups :  (i)  Those  aiming  to  quicken 
the  rate  of  mental  response.  (2)  Those  aiming  at 
short-cut  processes  of  calculation. 

Typical  of  the  first  are  (a)  the  use  of  an  es- 
tablished rhythm  as  the  child  attacks  a  column 
of  additions ;  (b}  the  device  of  having  children 
race  for  quick  answers,  letting  them  raise  their 
hands  or  stand  when  they  have  gotten  the  an- 
swer ;  (c)  the  assignment  of  a  series  of  problems 
for  written  work  under  the  pressure  of  a  restricted 
time  allotment  for  the  performance  of  each. 
These  and  similar  devices  are  much  used  in  the 
schools.  They  are  open  to  the  objection  that  they 
quicken  the  rate  of  the  better  students,  but  foster 
81 


TEACHING  PRIMARY  ARITHMETIC 

confusion,  error,  and  discouragement  among  the 
less  able  children,  thus  actually  hindering  speed. 
The  various  shorter  methods  which  represent 
an  effort  to  reduce  the  number  of  mental  pro- 
cesses required  are  usually  not  of  general  appli- 
cability, and  consequently  have  not  attained  gen- 
eral currency  in  the  elementary  schools  which  aim 
to  teach  merely  one  generally  available  and  ef- 
fective method  even  though  it  requires  more 
time,  special  expertness  being  left  to  later  de- 
velopment in  the  special  school  or  business  which 
requires  it. 

The  Relation  of  Accuracy  to  Speed 

It  has  come  to  be  quite  a  common  opinion 
among  teachers  that  the  fundamental  element  in 
rapid  arithmetical  work  is  certain  and  accurate 
calculation.  If  pupils  know  their  tables  of  com- 
binations and  are  sure  of  each  detail  of  calcula- 
tion, there  is  no  confusion  or  hesitancy ;  speed 
then  follows  as  a  matter  of  course.  This  belief, 
as  much  as  anything  else,  explains  why  the  lower 
schools  have  developed  few  special  means  for  at- 
taining speed  other  than  those  mentioned. 


THE  USE  OF  SPECIAL  ALGORISMS,  ORAL 
FORMS,   AND  WRITTEN   ARRANGEMENTS 

THE  methods  of  teaching  arithmetic  are  influ- 
enced not  only  by  the  aims  of  such  instruction, 
but  by  the  peculiar  nature  of  the  matter  taught. 
The  use  of  special  algorisms,  temporary  algor- 
istic  aids  or  teaching  "  crutches,"  oral  and  written 
forms  of  analysis,  are  of  considerable  moment 
in  determining  the  difficulties  and  therefore  the 
methods  of  teaching.  Their  condition  and  influ- 
ence will  need  to  be  given  some  slight  notice. 

The  Traditional  Nature  of  Algorisms  and  Forms 

The  algorisms  and  forms  used  in  the  American 
schools  are  those  that  have  been  determined  by 
social  and  educational  traditions.  It  is  probable 
that  wide  social  practice  has  largely  determined 
the  traditions,  though  it  must  be  admitted  that 
the  traditions  of  text-book  makers  have  also 
given  it  form.  In  consequence  the  ruling  school 
83 


TEACHING  PRIMARY  ARITHMETIC 

tradition  in  the  matter  of  algorisms  does  not  al- 
ways coincide  with  current  community  practice. 
It  is  probable  that  the  various  modes  of  comput- 
ing interest,  given  by  the  average  arithmetic  of 
ten  years  ago,  were  once  current  in  the  business 
world.  These  methods  have  changed  somewhat, 
and  the  school  form  of  computation  has  not  al- 
ways been  changed  to  accord.  Such  misadjust- 
ments  between  the  forms  used  in  school  and 
those  used  in  daily  life  are  not  numerous,  but 
they  are  more  frequent  than  they  ought  to  be. 

The  Number  of  Algorisms  Used 

The  use  of  various  algorisms  for  a  single  pro- 
cess is  not  very  frequent.  There  is  a  fairly  gen- 
eral prevalence  of  a  single  algorism  for  a  single 
process  among  American  teachers.  Certain  strik- 
ing exceptions  are  to  be  noted  in  connection  with 
subtraction  and  division,  where  the  so-called 
"  Austrian "  methods  are  being  brought  into 
competition  with  the  traditional  modes  of  the 
American  school.  It  may  be  said,  however,  that 
even  when  two  distinct  algorisms  are  in  contem- 
poraneous use  in  a  school  system,  the  teachers 
84 


USE  OF  SPECIAL  FORMS 

are  usually  careful  to  employ  only  one  algorism 
with  a  given  child.  Even  when  a  child  moves  from 
a  school  using  one  kind  of  algorism  to  a  school 
using  another,  the  tendency  is  to  allow  him  to 
follow  his  own  method. 

Reform  in  the  Use  of  Algorisms 

The  tendency  toward  the  substitution,  dupli- 
cation, and  modification  of  existing  algorisms  is 
inconsiderable,  and  very  recent.  The  introduction 
of  the  "Austrian"  algorisms,  already  mentioned, 
is  perhaps  the  most  conspicuous  movement,  hav- 
ing a  very  wide  group  of  adherents.  There  are, 
however,  a  group  of  teachers  and  educational 
psychologists  who  are  attempting  to  refine  teach- 
ing methods  so  as  to  attain  a  greater  economy 
and  efficiency  in  the  learning  process.  These  are 
responsible  for  a  movement  toward  the  reform 
of  existing  algorisms.  The  movement  expresses 
itself  in  a  number  of  ways,  —  it  offers  new  forms, 
modifies  others,  and  aims  to  bring  a  larger  sim- 
ilarity and  consistency  into  algorisms  employed 
in  the  various  stages  of  the  same  process.  Its 
function  is  always  to  simplify  for  the  child  and 
85 


TEACHING  PRIMARY  ARITHMETIC 

thus  to  increase  the  practical  efficiency  and  the 
mental  economy  of  his  methods. 

The  Standard  of  Social  Usage 

One  of  the  standards  set,  is,  that  as  far  as  is 
consistent  with  economy,  the  algorism  employed 
with  greatest  frequency  in  social  life  is  to  be 
preferred.  If  people  add,  subtract,  and  multiply 
with  their  figures  arranged  above  and  below  each 
other  in  vertical  form  3 ,  then  the  vertical  form 

9 

is  to  be  preferred  to  the  horizontal  method 
6+3=9  so  largely  used  and  imposed  by  text- 
book writers. 

The  Extended  Use  of  Acquired  Forms 

To  learn  two  forms  for  one  thing,  particularly 
when  one  has  no  sanction  either  in  current  use 
or  on  general  grounds  of  psychological  efficiency, 
is  a  waste.  Hence  there  is  an  increasing  disposi- 
tion both  in  general  practice  and  among  the 
more  critical  to  utilize  a  single  form  in  as  many 
places  as  possible.  "  Subtracting  by  adding "  is 
merely  using  the  same  association  and  word  form 
for  both  addition  and  subtraction.  Hence  only 
86 


USE  OF  SPECIAL  FORMS 

one  set  of  tables,  instead  of  two,  has  to  be  learned. 
The  meaning,  the  applicability,  and  the  visual 
form  of  addition  and  subtraction  are  still  different. 
Only  the  process  of  remembering  and  using  the 
fundamental  combinations  is  the  same.  But  this 
is  a  large  saving :  "  Dividing  by  multiplying  "  is 
an  analogous  situation,  though  not  so  much  em- 
ployed in  American  schools  as  "  subtracting  by 
adding."  The  most  radical  suggestion  for  utilizing 
a  simplified  common  form  is  one  in  which  these 
forms  of  division,  as  used  in  the  tables  18-7-6=3, 
in  short  division  6)1832,  and  in  long  division 


62)i8325(,  are  reduced  to  one  consistent  form 
in  all  three  cases,  as  6  [78,  6|  1832,  62  |  18325. 
Such  a  simplification  is  urged  in  other  situations. 
The  movement  has  not  passed  far  beyond  theo- 
retic acceptance,  though  several  city  school  sys- 
tems are  trying  the  experiment,  San  Francisco 
being  a  notable  instance. 

The  Use  of  "  Crutches"  or  Temporary  Algorisms 

The  use  of  special  and  temporary  algoristic 
aids  or  learning  "  crutches  "  in  mathematical  cal- 
culation is  one  of  the  problems  of  method  under 
87 


r  TEACHING  PRIMARY  ARITHMETIC 

constant  controversy.  Teachers  seem  fairly  evenly 
divided  upon  the  question.  Typical  situations  in 
which  such  "  crutches  "  are  used  may  be  noted 
as  follows  :  Changing  the  figures  of  the  upper 
number  in  "  borrowing  "  in  subtraction  ;  rewrit- 
ing figures  in  adding  and  subtracting  fractions. 
In  the  broad  sense  any  algorism  which  is  used 
during  the  teaching  or  learning  process  tempor- 
arily, to  be  abandoned  completely  later,  is  an 
"accessory  algorism"  or  "crutch."  The  objec- 
tions to  their  use  lie  in  the  facts  :  (i)  that  skill  in 
manipulation  is  learned  in  connection  with  stages 
and  forms  not  characteristic  of  final  practical 
use ;  (2)  that  this  implies,  psychologically  at  any 
rate,  the  waste  of  learning  two  forms  or  usages 
instead  of  one ;  and,  (3)  that  it  decreases  the 
speed  with  which  mathematical  calculation  is 
done.  If  there  is  a  drift  in  any  direction,  it  is  prob- 
ably toward  the  abandonment  of  "  crutches." 

Full  and  Short  Forms  of  Calculation 

The  division  of  opinion,  which  exists  in  con- 
nection with  the  temporary  use  of  special  algor- 
isms or  "crutches,"  likewise  exists  with  reference 
88 


USE  OF  SPECIAL  FORMS 

to  the  use  of  "  full  forms  "  and  "  short  forms  " 
of  manipulation  and  statement.  The  temporary 
use  of  a  "  full  form,"  in  a  case  where  a  "  short 
form  "  will  finally  be  used,  is  similar  to  the  em- 
ployment of  a  "crutch."  There  is  one  important 
difference,  however,  which  explains  the  relatively 
larger  presence  of  temporary  "  full  forms  "  than 
of  "crutches."  The  "full  form  "  is  an  accurate 
form  which  is  used  somewhere,  —  in  a  more  com- 
plex stage  of  the  same  process  or  in  some  other 
process  ;  the  "  crutch  "  is  not.  Thus  :  a  "  full 
form  "  in  column  addition  with  partial  totals  and 
a  final  total  of  partial  totals,  will  be  utilized  in 
column  multiplication,  the  "  long  division  form  " 
of  doing  "  short  division  "  (which  is  the  fully  ex- 
pressed form  of  dividing  by  a  number  of  one 
figure)  will  be  utilized  in  division  by  numbers  of 
more  than  one  figure. 

Forms  of  Analysis  or  Reasoning 

The  problem  of  form  applies  not  alone  to  the 

algorism  or  special  method  of  computation,  but 

likewise  to  the  special  methods  of  reasoning  used 

in  determining  the  specific  series  of  steps  to  be 

89 


TEACHING  PRIMARY  ARITHMETIC 

taken  in  achieving  the  answer.  In  every  problem 
the  child  solves,  he  must  not  only  decide  what  is 
to  be  done  (reason),  but  he  must  do  it  (calculate). 
There  are  forms  of  reasoning  as  there  are  forms 
of  calculation.  As  any  calculation  may  have 
several  algorisms,  the  solution  of  a  problem  may 
be  expressed  in  several  forms.  It  is  the  latter  dif- 
ficulty which  appears  in  the  teacher's  demands 
for  "  formal  analysis"  of  problems.  The  analysis 
is  usually  required  in  full  statement. 

The  Traditional  Requirement  of  Full  Formal 
Analysis 

It  has  been  a  very  general  requirement  in 
American  schools  that  the  child  give  a  full  oral  or 
written  statement  of  his  analysis  and  computation. 
Formal  statements  have  been  demanded  of  the 
child  as  much  on  the  side  of  reasoning  as  on  that 
of  calculation.  One  of  the  causes  of  this  demand 
is  found  in  the  tendency  of  the  teacher  to  en- 
courage full  statement  by  the  child,  merely  as 
a  revelation  of  his  inner  processes  so  that  the 
source  of  error  in  results  might  be  detected  and 
the  error  eliminated.  We  have  already  noted  this 
90 


USE  OF  SPECIAL  FORMS 

predisposition  of  the  teacher  to  call  for  full  oral 
and  written  statements  for  purposes  of  control  in 
the  various  methods  designed  to  achieve  and  safe- 
guard accuracy. 

The  Limitations  of  Full  Formal  Analysis 

There  is,  however,  a  marked  tendency  away 
from  formal  analysis  of  arithmetic  problems  in 
the  elementary  school,  just  as  there  is  a  move- 
ment away  from  a  formal  deductive  logic  in 
the  higher  schools.  Natural,  genetic  modes  of 
thought  are  supplanting  the  unnatural,  formal 
statement  of  steps.  It  is  felt  that  while  such  full 
formal  statements  of  reasoning  and  calculation 
may  assist  in  the  teacher's  control,  they  may 
actually  interfere  with  accuracy  and  rapidity  on 
the  child's  part. 

To  write  out  each  step  in  full  often  means 
giving  an  enlarged  attention  to  factors  that  are 
merely  touched  and  assumed  in  actual  thinking. 
To  delay  the  thought  process,  with  attention  held 
on  a  fully  developed  linguistic  statement,  whether 
oral  or  written,  may  be  to  distract  from  the  chain 
of  essential  ideas  or  meanings  that  really  solve 

91 


TEACHING  PRIMARY  ARITHMETIC 

the  problem.  Frequently  children  lose  the  thread 
of  thought  midway  of  the  process  because  of  the 
necessity  of  dealing  with  the  form  side,  and  have 
to  begin  anew. 

"  Labeling  "  the  Steps  of  Calculation 

A  conservative  protest  against  the  old  formal 
expression  of  reasoned  steps  is  found  in  omitting 
for  the  most  part  the  linguistic  statements  deal- 
ing with  the  logic  of  the  problem  and  merely 
"  labeling  "  the  numbers  that  occur  in  the  calcu- 
lation. This  is  a  more  restricted  form  of  state- 
ment, much  more  used  at  the  present  time  than 
hitherto.  But  it  is  still  open  to  psychological 
objections  that  make  the  more  scientific  critics 
protest.  There  are  many  stages  in  a  calculation 
where  there  is  no  association  whatever  with  the 
concrete  problem  in  hand.  The  concrete  problem 
is  studied,  the  decision  is  made  that  all  the  fac- 
tors named  are  to  be  added.  They  are  added, 
purely  abstractly,  and  a  number  is  given  as  the 
total.  The  result  is  then  thought  of  in  terms 
of  the  concrete  problem  in  hand.  A  disposition 
to  label  each  item  in  the  addition  may  be  ne- 
92 


USE  OF  SPECIAL  FORMS 

cessary  in  the  rendering  of  a  bill,  but  it  is  a 
false  and  obstructing  activity  in  the  actual  solv- 
ing of  the  problem.  The  same  situation  exists 
where  there  are  two  or  three  processes  to  be 
utilized  in  series.  Once  the  child  has  grasped  his 
concrete  situation  and  reasoned  what  to  do  —  he 
may  proceed  to  mechanical  manipulation,  never 
thinking  of  the  concrete  applications  till  he  has 
finished. 

So-called  Accuracies  of  Statement 

The  protest  mentioned  above  even  goes  so  far 
as  to  attack  the  teacher's  insistence  upon  certain 
so-called  accuracies  of  statement.  The  case  of  3 
pencils  at  5  cents  would  be  expressed  3  times  5 
cents  ==15  cents.  3x5  =  15  would  be  demanded, 
and  5x3  not  allowed  at  all.  The  protest  is  not 
against  insistence  on  a  proper  order  where  "  la- 
beled "  statements  are  used.  The  objection  is 
made  against  the  demand  for  the  so-called  proper 
order  when  abstract  figures  are  related  merely 
by  signs.  Where  the  child  calculates  ^symbol- 
ically, he  sees  the  situation  as  one  to  be  worked 
out  by  a  purely  conventional  relation  between 
93 


TEACHING  PRIMARY  ARITHMETIC 

two  numbers.  For  all  practical  purposes  5x3 
will  solve  the  situation  quite  as  accurately  as 
3x5.  The  insistence  on  one,  as  opposed  to  the 
other,  is  a  useless  effort  that  cannot  affect  the 
result. 

Increased  Use  of  Mathematical  Symbols 

The  same  tendency  which  is  making  for  a  re- 
duction of  verbal  forms  is  increasing  the  use  of 
mathematical  symbols.  As  logical  relations  are 
less  frequently  written  out,  a  simple  sign  such 
as  +  or  -f-  is  used.  The  algebraic  x  is  supplied 
in  place  of  a  whole  roundabout  series  of  awkward 
preliminary  statements  or  assumptions.  With  it, 
of  course,  come  changed  methods  of  manipula- 
tion, as  in  the  use  of  the  algebraic  equation. 

It  is  doubtless  true  that  the  rigidity  of  full 
logical  forms  is  giving  way  to  a  more  flexible  and 
natural  mode  of  expressing  the  child's  thoughts. 
Fixed  oral  and  written  forms  of  exposition  may 
assist  the  child,  much  as  the  acquisition  of  a 
definite  symbol  fixes  an  abstract  meaning,  which 
remains  unwieldy  till  it  attaches  itself  to  a  word 
by  which  it  is  to  be  recalled.  But  increasing  care 
94 


USE  OF  SPECIAL  FORMS 

is  manifested  that  children  shall  use  only  those 
forms  that  will  conform  to  practical  need  upon 
the  one  hand,  and  to  natural,  efficient,  and  eco- 
nomical mastery  on  the  other. 


XI 

EXAMPLES  AND  PROBLEMS 

Formal  and  Applied  Arithmetic 

THE  teaching  of  arithmetic  is  usually  classified 
under  two  aspects,  formal  work  and  applied  work. 
The  formal  work  deals  mainly  with  the  memori- 
zation of  fundamental  facts,  processes,  and  other 
details  of  manipulation.  The  applied  work,  as 
the  name  implies,  is  the  formal  work  utilized  in 
the  setting  of  a  concrete  situation  demanding  a 
solution.  These  two  aspects  of  arithmetical  in- 
struction are  very  frequently  sharply  separated, 
the  child  working  alternately  with  one  or  the 
other.  The  characteristic  practice  is  to  deal  with 
them  without  relating  them  as  closely  as  the 
highest  efficiency  would  demand. 

The  Example  and  the  Problem 

Formal  exercises  in  arithmetic  are  usually  pre- 
sented through  the  "  example  "  ;  the  exercises  in 
application  through  the  "  problem  " ;  the  distinc- 
96 


EXAMPLES  AND  PROBLEMS 

tion  being  that  one  is  an  abstract  and  symbolical 
statement  of  numerical  facts  and  the  other  a  con- 
crete and  descriptive  statement.1  In  the  first 
case  the  mathematical  sign  tells  the  child  what 
to  do,  whether  to  add,  subtract,  multiply,  or  di- 
vide ;  the  example  being  a  kind  of  pre-reasoned 
problem,  the  child  has  only  to  manipulate  ac- 
cording to  the  sign,  his  whole  attention  through- 
out being  focused  on  the  formal  calculation.  In 
the  second  case  the  child  has  two  distinct  func- 
tions ;  he  must,  from  the  description  of  the  sit- 
uation presented,  decide,  through  the  process  of 
reasoning,  what  he  is  to  do  (add,  subtract,  divide, 
or  multiply),  and  having  rendered  his  judgment, 
he  must  proceed  through  the  formal  calculation. 

The  Traditional  Precedence  of  Formal  Work 

As  the  problem  involves  two  types  of  mental 
processes  in  a  single  exercise,  and  the  example 

1  While  this  distinction  is  not  general,  it  has  sufficient  cur- 
rency to  warrant  its  use  here  for  the  convenience  of  discus- 
sion. The  expression  '*  clothed  problem  "  (from  the  German) 
is  occasionally  used  to  mean  what  is  here  designated  as 
"  problem,"  and  "  abstract  problem  "  is  used  to  mean  what  is 
here  designated  as  "  example." 

97 


TEACHING  PRIMARY  ARITHMETIC 

but  one,  the  usual  procedure  in  arithmetic  is  to 
take  up  the  formal  side  through  examples  first 
and,  later  on,  the  applied  side  through  the  use  of 
problems.  This  means  that  the  first  emphasis  is 
on  formal  and  abstract  work  rather  than  on  a 
treatment  of  natural,  concrete  situations,  an  em- 
phasis not  wholly  sanctioned  by  modern  psy- 
chology and  the  better  teaching  procedure  of 
other  subjects. 

Objective  and  Narrative  Presentation  as  a 
Reform  Tendency 

The  reform  tendency  is  found  mainly  in  the 
primary  grades  where  the  beginnings  of  new 
processes  are  made  through  objective  presenta- 
tions of  the  problem.  But  the  transition  from 
objectified  problems  to  formal  work  is  not  imme- 
diate. The  children  pass  from  objectified  situa- 
tions to  "number  stories,"  which  are  only  de- 
scriptions or  narratives  of  a  situation.  This  is 
the  interesting  primary-school  equivalent  of  that 
more  businesslike  language  description  found  in 
the  higher  grades,  the  arithmetical  problem.  But 
it  precedes  formal  work  and  succeeds  it,  —  the 
98 


EXAMPLES  AND  PROBLEMS 

formal  drill  being  a  mere  intermediate  drill. 
Here  concrete  presentations  and  formal  work  are 
more  closely  related  and  more  naturally  ordered. 
This  reform  tendency,  which  began  in  the  pri- 
mary school,  is  extending  to  the  higher  grades, 
where  it  is  no  longer  rare  to  find  the  attack  upon  a 
process  preceded  by  careful  studies  of  the  con- 
crete circumstances  in  which  the  process  is  util- 
ized. In  the  case  of  interest,  several  days  might 
be  utilized  in  studying  the  institution  of  banking 
in  all  its  more  important  facts  and  relations. 
Such  an  approach  not  only  provides  motive  for 
the  formal  and  mechanical  work,  but  gives  a 
necessary  logical  basis  in  fact.  Hence  the  un- 
derstanding of  practical  business  life  makes  ac- 
curate reasoning  possible  for  the  child  when  he 
is  called  upon  to  solve  actual  problems  in  appli- 
cation of  the  formal  work. 

The  Over-Emphasis  of  Formal  Work 

It  is  perfectly  natural  under  the  general  tradi- 
tional practice  of  putting  the  first  emphasis  on 
mastery  of  formal  work  that  the  largest  amount 
of  attention  should  be  given  to  the  mechanical 
99 


TEACHING  PRIMARY  ARITHMETIC 

and  technical  side  of  arithmetic,  and  that  the 
concrete  uses  and  applications  should  be  slighted, 
and  this  is  generally  true  of  the  practice  of 
American  teachers.  Much  more  ingenuity  has 
been  used  in  the  careful  training  of  the  child  on 
the  formal  side  than  in  teaching  him  to  think  out 
his  problems.  There  is  no  such  careful  arrang- 
ing and  ordering  of  types  in  teaching  a  child 
to  reason,  as  there  is  in  teaching  him  to  cal- 
culate. 

The  Need  for  More  Systematic  Teaching  of 
Reasoning 

Here  and  there  a  few  thoroughly  systematic 
attempts  are  made  to  carry  the  pupil  through 
the  simple  types  of  one-step  reasoning,  to  two- 
step  and  three-step  problems  with  their  possible 
variations.  Just  as  the  example  isolates  the  diffi- 
culties of  calculation,  by  letting  the  sign  +  or  — 
stand  for  the  logic  of  the  situation,  there  is  a 
tendency  to  give  problems  without  requiring  the 
calculations.  This  affords  a  means  of  isolating 
and  treating  the  special  difficulties  of  reasoning. 
The  child  is  merely  required  to  tell  what  he 
100 


EXAMPLES  AND  PROBLEMS 

would  do,  without  doing  it ;  the  answer  being 
checked  by  the  gross  facts.  A  little  later,  as  a 
transition,  he  is  permitted  to  give  a  rapid,  rough 
approximation  of  what  the  answer  is  likely  to  be. 
With  further  command  he  tells  what  he  would 
do  and  does  it  accurately.  But  such  a  program 
of  teaching  is  still  rare  among  teachers. 

Existing  Devices  for  Testing  Reasoning 

The  care  of  the  child's  reasoning  is  largely 
restricted  to  testing  his  comprehension  of  the 
problem,  (i)  by  having  him  restate  the  problem 
to  be  sure  he  understands  it,  or  (2)  by  having 
him  give  a  formal  oral  or  written  analysis  of  the 
way  in  which  he  solved  the  problem.  The  first 
requirement  may  not  be  thoroughgoing,  as  the 
child  may  give  a  verbal  repetition  of  the  problem 
without  really  knowing  its  meaning.  The  second 
is  a  formal  analysis  of  the  finished  result  and 
does  not  represent  the  genetic  method  of  the 
child's  thinking.  Consequently  its  formulas  do 
not,  in  any  considerable  degree,  assist  him  in  his 
actual  struggle  with  the  complex  of  facts. 


101 


TEACHING  PRIMARY  ARITHMETIC 

Sources  of  Failure  in  the  Solution  of  Problems 

This  lack  of  a  systematic  teaching  of  the  tech- 
nique of  reasoning  is  manifest  in  the  unrelia- 
bility of  children's  thinking.  When  a  child  fails 
in  a  problem  assigned  from  the  text-book,  the 
source  of  the  error  may  be  in  one  or  more  of 
three  phases  :  (i)  In  failing  to  get  the  meaning 
of  the  language  used  to  describe  the  details  of 
the  situation ;  (2)  in  failing  to  reason  out  what 
needs  to  be  done  to  solve  the  situation ;  (3)  in 
failing  to  make  an  accurate  calculation.  The 
first  is  a  matter  of  language  ;  the  second,  one  of 
reasoning ;  the  third,  one  of  memorization.  The 
elimination  of  errors,  due  to  the  first  and  third 
sources,  leaves  a  considerable  proportion  to  be 
accounted  for  by  the  second.  Such  informal  in- 
vestigations as  have  been  made  seem  to  show 
that  the  children  who  fail  in  reasoning  do  not 
make  any  real  effort  to  penetrate  into  the  essen- 
tial relations  of  the  situation.  They  depend  on 
their  association  of  processes  with  specific  words 
of  relation  used  in  the  description  of  the  prob- 
lem, an  association  determined  of  course  by  their 
1 02 


EXAMPLES  AND  PROBLEMS 

past  experiences.  As  long  as  these  familar  "  cue  " 
words  are  used,  they  succeed.  Let  unfamiliar 
words  or  phrases  be  utilized  in  their  stead  or  let 
the  relation  be  implied,  and,  like  as  not,  the 
children  will  fail  to  do  the  right  thing.  Practical 
school  people  are  familiar  with  the  fact  that 
children  solve  the  problems  given  in  the  lan- 
guage of  their  own  teachers  and  fail  when  the 
problems  are  set  by^grincipals  ^r_superintend- 
ents,  whose  language  is'  strange  to  them. 

/ 
The  Need  of  Varied  Presentations  of  Problems 

A  varied  use  of  materials  in  the  objective 
presentation  of  problems,  and  a  more  constantly 
varied  use  of  language  in  the  descriptive  pre- 
sentation of  problems  would  prevent  the  child 
from  making  such  superficial  and  unthought- 
ful  associations,  and  force  him  to  think  out 
connections  between  what  is  essential  in  a 
typical  problem  and  the  appropriate  process  of 
manipulating  it.  But  such  a  deliberate  applica- 
tion of  modern  psychology  is  far  from  being  a 
conspicuous  minority  movement.  The  subject- 
matter  of  the  problems  given  to  children  has, 
103 


TEACHING  PRIMARY  ARITHMETIC 

however,  improved  greatly.  Obsolete,  puzzling, 
and  unreal  situations,  which  only  hinder  the 
child's  attempt  to  reason,  are  less  and  less  used 
in  problem  work. 

Improvement  in  the  Subject- Matter  of  Problems 

Daily  it  becomes  recognized  with  greater  clear- 
ness that  right  reasoning  depends  upon  a  com- 
prehension of  the  facts  of  the  case,  and  the 
facts  of  the  case  in  point  must  be  within  the 
experience  of  the  child.  This  is  the  only  way 
in  which  a  problem  can  be  real  and  concrete 
to  him. 

Real  and  Concrete  Problems  Taken  from  the 
Larger  Social  World 

The  recent  effort  on  the  part  of  text-book 
writers  and  teachers  to  make  arithmetical  prob- 
lems "real"  and  "concrete"  has  not  always 
recognized  the  above-mentioned  psychological 
principle.  The  terms  "real"  and  "concrete" 
have  been  interpreted  in  many  ways  besides  in 
terms  of  the  child's  consciousness.  With  some, 
"  real"  has  meant  "  material"  ;  and  the  problems, 
104 


EXAMPLES  AND  PROBLEMS 

more  particularly  with  primary  children,  have, 
in  increasing  degree,  been  presented  by  objects 
or  words  connoting  very  vivid  images.  Others 
have  defined  this  quality  in  terms  of  actual  exist- 
ence or  use  in  the  larger  social  world.  If  these 
problems  actually  occur  at  the  grocer's,  the 
banker's,  or  the  wholesaler's,  it  is  said  that  they 
"are  indeed  concrete."  And  much  effort  has 
been  expended  in  carrying  these  current  prob- 
lems into  the  classroom,  in  spite  of  the  fact  that 
they  may  be  neither  comprehensible  nor  interest- 
ing to  the  pupil. 

Real  and  Concrete  Problems  Taken  from  the 
Child's  Own  Life 

There  is  another  social  world,  nearer  home  to 
the  child,  from  which  a  more  vital  borrowing  can 
be  made.  There  is  an  opportunity  to  use  the 
child's  life  in  its  quantitative  aspects,  to  take  his 
plays,  games,  and  occupations,  and  introduce 
their  situations  into  his  mathematical  teaching. 
As  his  world  expands  from  year  to  year,  he  will 
be  carried  by  degrees  from  personal  and  local 
situations  to  those  of  general  interest.  The 
105 


TEACHING  PRIMARY  ARITHMETIC 

teacher  can  provide  this  progression  without  de- 
vitalizing the  facts  presented. 

The  Imaginative  or  Hypothetical  Problem 

There  is  another  error  into  which  both  the 
socially-minded  radical  and  the  specialist  in  child 
study  fall.  In  their  eagerness  to  improve  the 
arithmetical  problem,  they  assume  that  problems 
taken  from  the  larger  sbciaDworld  or  from  the 
child's  experience  are  necessarily  superior  to  hy- 
pothetical, imaginative,  or  "  made-up  "  problems. 
The  psychological  fact  that  needs  to  be  forced 
upon  the  attention  of  the  reformers  is  that,  with 
proper  artfulness,  an  imagined  problem  may  be 
even  more  vital  and  real  to  the  child  than  one 
taken  from  life  —  as  a  situation  in  a  drama  may 
be  more  appealing  and  real  to  a  child  than  one  on 
the  street.  This  has  some  recognition,  but  not 
enough.  Those  who  stand  upon  the  side  of  the 
"  made-up  "  problems  are  more  likely  to  be  re- 
actionaries who  tolerate  the  traditional  type  of 
problem  even  though  its  stupid  artificiality  is  ob- 
vious to  both  the  teacher  and  the  child.  They 
might  better  be  dealing  with  dull  problems  bor- 
106 


EXAMPLES  AND  PROBLEMS 

rowed  from  real  life  than  with  specially  invented 
dullness. 

Valid  Arguments  for  Actual  Problems 

Of  course  there  is  another  argument  for  the 
use  of  actual  social  materials.  The  child  must 
ultimately  come  into  command  of  precisely  these 
facts,  since  their  mastery  will  be  demanded  by 
the  business  world.  But  must  a  primary  school 
child  study  his  arithmetic  through  problems 
taken  from  the  dreary  statistics  of  imports  and 
exports  merely  because  tariff  reform  is  a  polit- 
ical issue  which  every  citizen  ought  finally  to 
comprehend  ?  There  is  a  time  for  this,  and,  as 
is  the  case  with  most  of  such  borrowed  busi- 
ness problems,  the  time  is  later.  In  so  far  as 
these  are  current  situations  within  the  con- 
tacts of  child  life,  let  them  enter.  A  quan- 
titative revelation  of  life  is  important ;  and  it 
is  good  teaching  economy  to  gain  knowledge 
by  the  way,  provided  it  does  not  distract 
attention  from  whatever  main  business  is  at 
hand. 


107 


TEACHING  PRIMARY  ARITHMETIC 

Unity  in  the  Subject-Matter  of  Problems 

The  socializing  of  arithmetical  problems  has 
one  other  additional  good  effect.  It  has  tended 
to  bring  some  topical  unity  into  the  problems 
constituting  the  assignment  for  a  given  lesson, 
or  group  of  lessons.  Hitherto,  a  series  of  prob- 
lems was  almost  always  composed  of  a  hetero- 
geneous lot  of  situations.  There  was  no  unity 
save  that  some  one  process  was  involved  in  each. 
The  movement  is  now  in  the  direction  of  attain- 
ing a  more  approximate  unity  within  the  subject- 
matter  of  the  problems  themselves.  The  difficul- 
ties of  attainment  have  restricted  this  movement 
to  more  progressive  circles. 

The  Eclectic  Source  of  Problems 

The  eclectic  source  of  arithmetic  problems  is 
apparent  from  the  foregoing  discussion.  It  would 
seem  that  some  better  texts  would  naturally  be 
evolved  through  the  implied  criticism  of  each 
movement  upon  the  other.  Such  is  the  case. 
Problems  from  child  life  emphasize  the  begin- 
ning condition  to  which  adjustment  must  be 
1 08 


EXAMPLES  AND  PROBLEMS 

made  in  all  good  teaching.  Those  from  the 
greater  world  suggest  the  final  goals  of  instruc- 
tion. Those  "made  up"  by  the  teacher  call 
attention  to  what  is  too  often  forgotten,  that  the 
educative  process  in  school  may  be  artful  with- 
out becoming  artificial.  Teaching  is  art,  and 
when  well  done  is  not  less  effective  for  the  fact. 


XII 

CHARACTERISTIC  MODES  OF   PROGRESS  IN 
TEACHING    METHOD 

Variation  in  Method,  and  its  Causes 

THE  existing  methods  of  teaching  arithmetic  in 
the  American  elementary  schools  are  exceed- 
ingly varied.  This  is  due  to  many  causes.  The 
democratic  system  of  local  control,  as  opposed 
to  a  centralized  supervision  of  schools,  has  in- 
creased both  the  possibility  and  the  probability  of 
variation.  Even  within  the  units  of  supervision 
(state,  county,  and  municipal)  the  opportunity 
for  reducing  variation  in  the  direction  of  a  more 
efficient  uniformity  is  lost.  This  is  partly  due  to 
the  lack  of  a  thoroughly  trained  staff  of  super- 
visors of  the  teaching  process.  Uniformity  be- 
yond the  legal  units  of  supervision  has  been 
restricted  by  the  lack  of  organized  professional 
means  for  investigation  of  and  experimentation 
in  controversial  matters.  Even  such  crude  ex- 
no 


PROGRESS  IN  TEACHING  METHOD 

periments  as  are  being  tried  in  more  than  one 
class  room,  school,  or  system  are  unknown,  un- 
reported,  unestimated,  because  no  competent 
professional  body  gathers,  evaluates,  and  diffuses 
such  knowledge.  In  this  respect  the  teaching 
profession  is  far  below  the  efficient  organization 
of  the  legal  and  medical  professions. 

Characteristic  Traditions  and  Reforms 

It  is  exceedingly  difficult  therefore  to  analyze 
the  characteristic  aspects  of  teaching  method 
except  as  these  are  interpreted  in  movements  of 
general  significance.  These  may  be  actual  or 
potential,  traditional  or  reformatory,  general  or 
local  in  present  acceptance.  The  situation  is  one 
wherein  tradition  is  mixed  with  radicalism,  and 
radicalism  modified  by  reaction.  In  this  medley 
of  movements  there  are  dominant  tendencies 
both  traditional  and  progressive. 

Forces  for  Progress  in  Method 

It  is  quite  impossible  to  indicate  the  progres- 
sive tendencies  with  clearness  save  in  connection 
with  the  discussions  of  concrete  difficulties  in 
in 


TEACHING  PRIMARY  ARITHMETIC 

mathematical  teaching.  The  forces  that  are  be- 
hind these  tendencies  may,  however,  be  summa- 
rized here.  For  convenience,  they  may  be  classi- 
fied into  eight  types  of  influence,  extending  from 
more  or  less  vague  and  general  movements  to 
very  particular,  scientific  contributions.  No  at- 
tempt is  made  to  indicate  the  achievement  of 
each  ;  the  form  of  each  influence  is  only  roughly 
defined,  and  illustrative  movements  or  studies 
are  suggested :  — 

General  Pedagogical  Movements 

(i)  It  is  obvious  that  any  general  pedagogical 
movement  that  influences  the  professional  atti- 
tude of  teachers  will  influence  the  special  meth- 
ods of  mathematical  teaching.  The  appearance 
of  the  doctrine  of  interest  made  mathematical 
instruction  less  formal.  The  growing  enthusi- 
asm for  objective  work  enlarged  the  use  of  ob- 
jects in  the  arithmetic  period.  The  child  study 
movement  laid  emphasis  upon  the  child's  own 
plays  and  games  as  a  source  of  problems  and  ex- 
amples. 


112 


PROGRESS  IN  TEACHING  METHOD 

Special  Pedagogical  Movements 

(2)  Certain  special  movements  in  methods  of 
teaching,  local  to  the  subject  of  mathematics, 
have  also  been  effective.    Here  one  has  only  to 
recall  the  "Grube"  method,  with  its  influence 
on  the  order  and  thoroughness  with  which  the 
elements  of  arithmetic  are  taught. 

Daily  Trial  and  Error 

(3)  The  tendency  of  every  teacher,  who  is  at 
all  sensitive  to  the  defects  of  his  methods,  is  to 
vary  his  daily  practice.  Constant  trial,  with  error 
eliminating  and  success  justifying  a  departure,  is 
thus  a  source  of  progress.   The  new  devices  of 
one  teacher  are  taken  up  by  the  eager  profes- 
sional witness,  and  innovation  is  thus  diffused. 
We  have  no  ability  to  measure  how  much  pro- 
fessional progress  is  due  to  individual  variation 
in  teaching  and  its  conscious  and  unconscious 
imitation.   The  disposition  of  school  systems  to 
send  their  teachers  on  tours  of  visitation  without 
loss  of  salary  is  a  recognition  of  the  value  of 
this  method  of  advance. 


TEACHING  PRIMARY  ARITHMETIC 

Experimentation  of  Progressive  Teachers 

(4)  A  far  more  efficient  and  radical  source  of 
change  than  that  just  mentioned  is  the  delib- 
erate, conscious,  experimental  teaching  of  pro- 
gressive individuals.   Some  new  idea  or  device 
occurs  to  the  teacher  of  original  mind,  and  it  is 
tried  out  with  a  fair  proportion  of  resulting  suc- 
cesses.   An  illustration  of  such  a  contribution  is 
found  in  one  conspicuous  effort  to  get  more  rapid 
column  addition.    The  first  columns  to  be  added 
were  allowed  to  determine  the  selection   and 
order  of  addition  combinations  learned.    Thus  if 
6  +  7  +  9  +  6  +  7=35  is  the  first  column  to  be  used, 
then  the  first   combinations   mastered  will  be 
6  +  7=13,  3  +  9=12,  2  +  6  =  8,  8  +  7=15.   Arising 
as  a  fruitful  idea  and  seeming  to  give  a  measure 
of  success,  it  has  been  carried,  in  the  particular 
locality  in  mind,  from  school  to  school,  and  from 
system  to  system. 

Reconstruction  through  Psychological  Criticism 

(5)  A  prolific  source  of  radical  change  is  found 
in  the  critical  application  of  modern  psychology 

114 


PROGRESS  IN  TEACHING  METHOD 

to  teaching  methods.  Algorisms,  types  of  diffi- 
culty, the  order  and  gradation  of  these,  as  well 
as  many  other  factors  in  method  have  been  rad- 
ically reorganized  on  psychological  grounds.  Ex- 
amples of  such  psychological  modifications  of 
method  are  found  in  the  "  Courses  of  Study  for 
the  Day  Elementary  Schools  of  the  City  of  San 
Francisco."  Still  more  extensive  critical  applica- 
tions are  found  in  the  "  Exercises  in  Arithmetic  " 
devised  by  Dr.  E.  L.  Thorndike,  Professor  of 
Educational  Psychology  in  Teachers  College, 
Columbia  University. 

Studies  in  the  Special  Psychology  of 
Mathematics 

(6)  Attempts  have  been  made  to  inquire  into 
the  special  psychology  of  arithmetical  processes 
through  careful  experimentation  and  control. 
They  have  not  been  numerous,  nor  have  they 
been  influential  on  current  practice.  Such  a  field 
needs  development.  A  typical  attempt  to  investi- 
gate and  formulate  the  special  psychology  of 
number  is  found  in  a  Clark  University  study  of 


TEACHING  PRIMARY  ARITHMETIC 

"Number  and  its  Application  Psychologically 
Considered."1 

Investigations  of  Existing  Methods 

(7)  Educational  investigations  as  to  the  effi- 
ciency of  existing  arithmetical  teaching  among 
school  systems,  sufficiently  varied  to  be  repre- 
sentative of  American  practice,  have  also  been 
conducted.  These  have  usually  gone  beyond  the 
field  of  the  special  methods  of  presentation  em- 
ployed in  the  classroom,  and  have  inquired  into 
the  conditions  of  administration  and  supervision, 
the  arrangement  of  the  courses  of  study,  and 
other  similar  factors.  Dr.  J.  M.  Rice's  studies  into 
"The  Causes  of  Success  and  Failure  in  Arith- 
metic"2 investigated  such  specific  factors  as: 
The  environment  from  which  children  come,  their 
age,  time  allotment  of  the  subject,  period  of  school 
day  given  to  arithmetic,  arrangement  of  home 
work,  standards,  examinations,  etc.  A  subsequent 

1  Phillips,  D.  E.,  "  Number  and  its  Application  Psycholog- 
ically   Considered,"    Pedagogical  Seminary,   1897-8,    voL  5, 
pp.  221-281. 

2  Rice,  J.  M.,  "  Educational  Research  :  Causes  of  Success 
and  Failure  in  Schools,"  Forum,  1902-03,  vol.  34,  pp.  281-97, 
437-S2- 

116 


PROGRESS  IN  TEACHING  METHOD 

study  of  similar  type,  but  employing  more  refined 
methods,  is  that  of  Dr.  C.  W.  Stone  on  "Arith- 
metical Abilities  and  some  Factors  determining 
them." 1  The  main  problem  of  this  study  was  to 
find  the  correlation  between  types  of  arithmet- 
ical ability  and  different  time  expenditures  and 
courses  of  study.  These  two  studies  have  prob- 
ably attracted  more  general  notice  than  any  other 
studies  of  arithmetical  instruction.  While  they 
have  largely  dealt  with  administrative  conditions 
that  limit  teaching  method,  rather  than  with  the 
details  of  teaching  method  itself,  they  have  stim- 
ulated the  impulse  to  investigate  conditions  and 
practices  of  every  type. 

Special  Experiments  in  Controlled  Comparative 

Teaching 

(8)  The  latest  source  of  progress  in  teaching 
method  is  found  in  the  movement  for  compara- 
tive experimental  teaching  under  normal  but  care- 
fully controlled  conditions.  Several  such  exper- 
iments are  being  conducted  in  the  Horace  Mann 

1  Stone,  C.  W.,  "  Arithmetical  Abilities  and  Some  Factors 
determining  them,"  Columbia  University  Contributions  to  Ed- 
ucation, Teachers  College,  N.  Y.  City,  1909,  p.  101. 

II/ 


TEACHING  PRIMARY  ARITHM  TIC 

Elementary  School  of  Teachers  College,  Colum- 
bia University,  under  the  direction  of  Principal 
Henry  C.  Pearson,  with  the  co-operation  of  the 
staff  of  Teachers  College.  This  experimental 
work  is  designed  to  determine  primarily  the 
relative  value  of  competing  methods  in  actual 
use  throughout  the  country,  the  assumption  being 
that  every  substantial  difference  in  practice  im- 
plies a  difference  of  theory  and  consequently  a 
controversy  that  can  be  resolved  only  on  the 
basis  of  careful  comparative  tests.  Two  parallel 
series  of  classes  of  about  the  same  age,  ability, 
teacher  equipment,  etc.,  are  selected  for  this 
work.  One  series  is  taught  by  one  method  ;  the 
other  series  by  the  other  method.  The  abilities 
of  these  children  are  measured  both  before  and 
after  the  teaching,  and  the  growth  compared. 
The  standards  and  methods  of  this  type  of  com- 
parative experimentation,  together  with  a  list  of 
current  competitive  methods  requiring  investiga- 
tion, are  given  in  Dr.  David  Eugene  Smith's 
monograph  on  "  The  Teaching  of  Arithmetic." 1 

1  Smith,  D.  E.,  "  The  Teaching  of  Arithmetic,"  chap,  xvi, 
Teachers  College,  January,  1909. 


OUTLINE 

I.    THE  SCOPE  OF  THE  STUDY 

1.  Function  of  the  Study  to  Trace  General  Tenden- 

cies    I 

2.  Teaching  Method  is  a  Mode  of  Presentation  .    .  2 

3.  Distinct  Uniformities  Exist  among  its  Variations  4 

4.  The  Methods  of  Public  Elementary  Schools  are 

Representative 4 

5.  Elementary  Mathematics  is  Mainly  Arithmetic    .  5 

6.  Elementary   Arithmetic   Emphasizes    the    Four 

Fundamental  Processes 6 

7.  The    Need  for    Studying   Exceptional   Reform 

Tendencies 8 

I.    THE  INFLUENCE  OF  AIMS  ON  TEACHING 

1.  Factors  Influencing  Teaching  Methods  ....    9 

2.  The  Influence  of  a  Scientific  Aim 9 

3.  The  Influence  of  the  Aim  of  Formal  Discipline  .  12 

4.  The  Shift  in  Emphasis  from  Academic  to  Social 

Aims 14 

5.  Business  Utility  as  an  End 15 

6.  Broad  Social  Utilitarianism  as  a  Standard  .    .    .  17 

7.  Some  Concrete  Effects  of  the  Change  in  Aim    .  19 

119 


OUTLINE 

m.    THE  EFFECT  OF  THE  CHANGING  STATUS 
OF  TEACHING  METHOD 

1.  Method  as  Psychological  Adjustment  to  the  Child  21 

2.  The  Effect  of  an  Increased  Reverence  for  Child- 

hood      22 

3.  The  Reconstruction  of  Method  through  Psychol- 

ogy   23 

4.  The  Increased  Professional  Respectability  of  Con- 

scious Method 25 

5.  The  Prevalence  of  Methods  Emphasizing  a  Single 

Idea 26 

6.  The  Tendency  toward  Over-Uniformity  in  Method  28 

7.  Method  as  a  Series  of  Varied,  Particular  Adjust- 

ments   30 

IV.  METHOD  AS  AFFECTED  BY  THE  DIS- 
TRIBUTION AND  ARRANGEMENT  OF 
ARITHMETICAL  WORK 

1.  The  Tendency  toward  Shortening  the  Time  Dis- 

tribution   32 

2.  The  Attempt  to  Eliminate  Waste 33 

3.  Delay  in  Beginning  Formal  Arithmetic  Teaching  34 

4.  The  Incidental  Method  of  Teaching 35 

5.  Reactions  against  the  Plan  of  Incidental  Teaching  36 
j  6.   Logical  and  Psychological  Types  of  Arrangement  38 

7.  Estimates  of  Worth 40 

8.  The  Present  Mixed  Method  of  Procedure  ...  41 

120 


OUTLINE 

V.  THE  DISTRIBUTION  OF  OBJECTIVE  WORK 

1.  Objective  Teaching  is  Generally  Current    ...  42 

2.  Its  Distribution  is  Crudely  Gauged 43 

3.  Tendency  toward  a  More  Refined   Correlation 

of  Object-Teaching  with  Particular  Immatu- 
rity   44 

4.  The  Movement  Supported  by  both  Scientific  and 

Common-Sense  Criticism 46 

VI.  THE  MATERIALS  OF  OBJECTIVE  TEACHING 

;  I.  The  Indiscriminate  Use  of  Objects    .    .    .    .    .47 

2.  The  Artificiality  of  Materials  Utilized    ....  47 

3.  Narrowness  in  the  Range  of  Materials    ....  48 

4.  Inadequate  Variation  of  Traditional  Materials     .  49 

5.  The  Restricted  Use  of  Diagrams  and  Pictures    .  50 

6.  Plays  and  Games  in  Object  Teaching     .    .     .    .51 

7.  The  Lack  of  Unity  in  the  Use  of  Objects   .    .    .52 

VII.  SOME  RECENT  INFLUENCES  ON   OBJEC- 

TIVE TEACHING 

1.  The  Influence  of  Inductive  Teaching 53 

2.  The  Movement  for  Active  Modes  of  Learning     .  55 

3.  The  Abbreviated  Use  of  Objects 57 

4.  The  Method  of  Parallel  Correspondence     ...  57 

5.  The  Method  of  Final  Correspondence    ....  58 


121 


OUTLINE 

Vffl.     THE  USE  OF  METHODS  OF  RATIONALI- 
ZATION 

1.  The  Tendency  toward  Rational  Methods  ...  60 

2.  The  Era  of  Direct  Instruction  and  Drill     ...  60 

3.  Indirect  Teaching  as  a  Rational  Method    ...  62 

4.  Interest  as  a  Factor  in  Methods  of  Rationalization  63 

5.  The  Reaction  against  Rationalization      ....  64 

6.  Four  Principles  for  the  Use  of  Rationalization    .  65 

7.  The  Substantiating  Psychology 67 

8.  Rationalization  as  a  Substitute  for  Object  Teach- 

ing   67 

IX.     SPECIAL  METHODS  FOR  OBTAINING  AC- 
CURACY, INDEPENDENCE,  AND  SPEED 

1.  Supervision  of  Learning  after  First  Development  69 

2.  The  Use  of  Steps,  or  Stages,  in  Teaching  ...  70 

3.  Stages  in  the  Presentation  of  Problems  .     .    .    .71 

4.  An  Opposite  Method  in  Presenting  Examples     .  73 

5.  Better  Transitions   from   Concrete  to   Abstract 

Work -73 

6.  The  Child's  Four  Modes  of  Work 75 

7.  The  Worth  of  these  Modes 76 

8.  The  Traditional  Quarrel  between  "  Mental "  and 

"Written"  Arithmetic 77 

9.  The  Transition  from  Development  by  Teacher  to 

Independent  Work  by  Pupil 79 

10.  Four  Characteristic  Stages  of  the  Transition  .    .  79 
122 


OUTLINE 

11.  Special  Methods  of  Attaining  Speed 81 

12.  The  Relation  of  Accuracy  to  Speed 82 

X.    THE    USE    OF    SPECIAL  ALGORISMS,   ORAL 
FORMS,  AND  WRITTEN  ARRANGEMENTS 

1.  The  Traditional  Nature  of  Algorisms  and  Forms  83 

2.  The  Number  of  Algorisms  Used 84 

3.  Reform  in  the  Use  of  Algorisms 85 

4.  The  Standard  of  Social  Usage 86 

5.  The  Extended  Use  of  Acquired  Forms  ....  86 

6.  The  Use  of  "Crutches  "  or  Temporary  Algorisms  87 

7.  Full  and  Short  Forms  of  Calculation 88 

8.  Forms  of  Analysis  or  Reasoning 89 

9.  The   Traditional   Requirement  of  Full   Formal 

Analysis 90 

10.  The  Limitations  of  Full  Formal  Analysis    ...  91 
ir.   "  Labeling  "  the  Steps  of  Calculation 92 

12.  So-called  Accuracies  of  Statement 93 

13.  Increased  Use  of  Mathematical  Symbols    ...  94 

XI.    EXAMPLES  AND    PROBLEMS 

1.  Formal  and  Applied  Arithmetic 96 

2.  The  Example  and  the  Problem 96 

3.  The  Traditional  Precedence  of  Formal  Work  .     .  97 

4.  Objective  and  Narrative  Presentation  as  a  Re- 

form Tendency 98 

5.  The  Over-Emphasis  of  Formal  Work    ....    99 

6.  The  Need  for  More  Systematic  Teaching  of  Rea- 

soning     ico 

123 


OUTLINE 

7.  Existing  Devices  for  Testing  Reasoning  .    .     .  101 

8.  Sources  of  Failure  in  the  Solution  of  Problems    .  102 

9.  The  Need  of  Varied  Presentations  of  Problem    .  103 

10.  Improvement  in  the  Subject-Matter  of  Problems  104 

11.  Real  and  Concrete  Problems  Taken  from  the 

Larger  Social  World 104 

12.  Real  and  Concrete  Problems  Taken  from  the 

Child's  Own  Life 105 

13.  The  Imaginative  or  Hypothetical  Problem    .    .  106 

14.  Valid  Arguments  for  Actual  Problems  ....  107 

15.  Unity  in  the  Subject-Matter  of  Problems  .    .    .  108 

16.  The  Eclectic  Source  of  Problems 108 

XH.    CHARACTERISTIC  MODES  OF  PROGRESS 
IN  TEACHING  METHOD 

1.  Variation  in  Method  and  its  Causes no 

2.  Characteristic  Traditions  and  Reforms  .    .    .    .  ill 

3.  Forces  for  Progress  in  Method 112 

4.  General  Pedagogical  Movements 112 

5.  Special  Pedagogical  Movements 113 

6.  Daily  Trial  and  Error 113 

7.  Experimentation  of  Progressive  Teachers     .    .114 

8.  Reconstruction  through  Psychological  Criticism  114 

9.  Studies  in  the  Special  Psychology  of  Mathe- 

matics  115 

10.  Investigations  of  Existing  Methods 1 16 

n.  Special  Experiments  in  Controlled  Comparative 

Teaching 117 


OUTLINE 


7.  Existing  Devices  for  Testing  Reasoning  .    .    .  101 

8.  Sources  of  Failure  in  the  Solution  of  Problems    .  102 

9.  The  Need  of  Varied  Presentations  of  Problem    .103 

10.  Improvement  in  the  Subject-Matter  of  Problems  104 

11.  Real  and  Concrete  Problems  Taken  from  the 

Larger  Social  World    .........  104 

12.  Real  and  Concrete  Problems  Taken  from  the 

Child's  Own  Life     ..........  105 

13.  The  Imaginative  or  Hypothetical  Problem    .    .  106 

14.  Valid  Arguments  for  Actual  Problems  ....  107 

15.  Unity  in  thf»  Suhiprt-MaH-Ar  nf  Pr/->Klemc  T«« 

16.  TheE 


xn.  CH. 

1.  Variati 

2.  Charac 

3.  Forces 

4.  Gener; 

5.  Specia 

6.  Daily  ' 

7.  Experi 

8.  Recons 


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matics 115 

10.  Investigations  of  Existing  Methods 116 

11.  Special  Experiments  in  Controlled  Comparative 

Teaching 117 


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